Advanced Topics · Chapter 41

Logarithms

Logarithms turn powers into multiplication and products into sums. Learn the definition, the log rules and the base change formula that unlock AMC 12 and AIME log problems (this topic is mainly relevant for the AMC 12).

What's this chapter about?

41.1 Basic Definitions
41.2 Logarithmic Formulas
41.3 Practice Problems

You'll need

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41.10%

Basic Definitions

In this section you will be able to read a logarithm as a question about powers, and switch between log form and exponent form.

A log is just a question: “to what power?” Two to what power is 8? Three. So $\log_2 8 = 3$ .

Every logarithm hides an exponent equation. $\log_b n = p$ says exactly the same thing as $b^{p} = n$ . When a log problem looks scary, rewrite it in exponent form. When an exponent problem looks stuck, take a log.

Look: the widget’s top line counts steps (exponents). The bottom line shows where those steps land. Equal steps up top mean multiplying by $b$ down below.

Exploreplay first, formulas later

Try this: Drag the value marker to 32. What exponent sits above it? So what is log₂ 32?
Try this: Move the exponent marker one step right at a time. What happens to the value each time?
Try this: Drag the value to 3. Why is log₂ 3 not a whole number? Between which two whole numbers is it?

Name it

Definition 41.1.1 · Logarithm

A logarithm is the power to which a number must be raised in order to get some other number.

Logarithms are written as

a = \log_b n

where $b$ is the base and $n$ is the number.

Basically, we are asking how many times we need to multiply the base to get the number $n$ : what power do we need to raise the base to, to get $n$ ?

Theorem 41.1.2 · Converting to Logarithm and Exponents

\log_x y = a \implies x^a = y

The arrow works both ways: $x^a = y$ also means $\log_x y = a$ .

Try it

Problem 41.3.1

AMC 12

What is the value of $a$ for which $\frac{1}{\log_2 a} + \frac{1}{\log_3 a} + \frac{1}{\log_4 a} = 1$ ?

Problem 41.3.3

AMC 12

What is the value of $\left(625^{\log_5 2015}\right)^{\frac{1}{4}}$ ?

Problem 41.3.16(additional)

AMC 10

Suppose that $4^a = 5$ , $5^b = 6$ , $6^c = 7$ , and $7^d = 8$ . What is $a \cdot b \cdot c \cdot d$ ?

Checkpoint for section 41.1

Get 2 of 3 right to master this section. One try each, no hints.

41.20%

Logarithmic Formulas

In this section you will be able to use the product, quotient, power and base change rules to simplify logs and solve log equations.

$\log_{3x} 4 = \log_{2x} 8$ . Two different bases, both with $x$ inside. Rewrite both in base 2 and it falls apart in three lines.

Logs turn multiplication into addition. That is why a slide rule works: lay the length $\log 2$ next to $\log 3$ and you land on $\log 6$ .

The base change formula is the big one. When the bases don’t match, move everything into one base you like (often a prime), then simplify. Always check the domain at the end: every log needs a positive input and a positive base other than 1.

Exploreplay first, formulas later

Try this: The sliding scale's 1 sits over 2. Where does 6 appear on the fixed scale? Why is it right under the sliding 3?
Try this: Press Power with a = 2 and k = 3. How many copies of log 2 make log 8?
Try this: Press Show me under the Base Change Formula. Why is every base-4 step exactly two base-2 steps?

Name it

Theorem 41.2.1 · Important Formulas

\log_a a^r = r

\log_a bc = \log_a b + \log_a c

\log_a \frac{b}{c} = \log_a b - \log_a c

\log_a b^c = c \log_a b

\log_a b \cdot \log_b c = \frac{\log b}{\log a} \cdot \frac{\log c}{\log b} = \frac{\log c}{\log a} = \log_a c

\log_b a = \frac{1}{\log_a b}

\log_b a = \frac{\log_d a}{\log_d b}

Remark 41.2.3

These formulas are extremely important for working with logarithms and should definitely be memorized.

Remark 41.2.4

If you ever forget which way the signs in these logarithm rules go, just try a small example like $\log_{10} 100 + \log_{10} 1000 = \log_{10} 100{,}000$ . From there you can figure out the sum-of-logarithms identity.

Remark 41.2.5

Natural logarithm (base $e$ ), where $e$ is Euler's number, $e \approx 2.71828$ .

A logarithm with base $e$ is called a natural log and is written $\ln$ .

Theorem 41.2.6 · Advanced Formulas

\log_{a^m} a^n = \frac{n}{m}

\log_a \left(\frac{1}{b}\right) = -\log_a b

\log_{\frac{1}{a}} b = -\log_a b

Concept 41.2.8

When you take the logarithms of numbers that form a geometric progression, the logarithms of those numbers form an arithmetic progression.

Watch itpredict, then reveal

Example 41.1

AMC 12

The solutions to the equation $\log_{3x} 4 = \log_{2x} 8$ , where $x$ is a positive real number other than $\frac{1}{3}$ or $\frac{1}{2}$ , can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$ ?

1
The bases $3x$ and $2x$ are messy, but $4$ and $8$ are both powers of $2$ . Change to base $2$ : $\log_{3x} 4 = \frac{\log_2 4}{\log_2 (3x)}$ . What is the numerator $\log_2 4$ ?

Predict first:
Your prediction

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Example 41.2

AIME

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$ , for some fixed $b \ge 2$ . A Martian student writes down

3 \log\left(\sqrt{x} \log x\right) = 56

\log_{\log x}(x) = 54

and finds that this system of equations has a single real number solution $x > 1$ . Find $b$ .

1
Every log here is base $b$ . Let $y = \log x$ . The second equation says $\log_y x = 54$ . Convert it to an exponent equation. What is $x$ in terms of $y$ ?

Predict first:
Your prediction

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Example 41.3

AIME

Let $x$ , $y$ , and $z$ be real numbers satisfying the system

\begin{aligned} \log_2(xyz - 3 + \log_5 x) &= 5, \\ \log_3(xyz - 3 + \log_5 y) &= 4, \\ \log_4(xyz - 3 + \log_5 z) &= 4. \end{aligned}

Find the value of $|\log_5 x| + |\log_5 y| + |\log_5 z|$ .

1
Convert each equation to exponent form. The first says $xyz - 3 + \log_5 x = 2^5$ . What is $\log_5 x + xyz$ ?

Predict first:
Your prediction

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Example 41.4

AIME

Let $x$ , $y$ , and $z$ be positive real numbers that satisfy

2\log_x(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \ne 0.

The value of $xy^5z$ can be expressed in the form $\frac{1}{2^{p/q}}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$ .

1
Everything is a power of $2$ times $x$ , $y$ or $z$ , so work in base $2$ . Let $u = \log_2 x$ , $v = \log_2 y$ , $w = \log_2 z$ , and call the common value $k$ . With the base change formula, what does $2\log_x(2y) = k$ become?

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Try it

Problem 41.3.2

AMC 12

What is the value of

\log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27?

Problem 41.3.23(additional)

AMC 12

If $\log\left(xy^3\right) = 1$ and $\log\left(x^2y\right) = 1$ , what is $\log(xy)$ ?

Problem 41.3.25(additional)

AMC 12

The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \log_b 7^7$ . What is $b$ ?

Contest cornerreal contest problems

Problem 41.3.4

AMC 12

There is a unique positive integer $n$ such that

\log_2(\log_{16} n) = \log_4(\log_4 n).

What is the sum of the digits of $n$ ?

Problem 41.3.6

AMC 12

The sequence

\log_{12} 162,\ \log_{12} x,\ \log_{12} y,\ \log_{12} z,\ \log_{12} 1250

is an arithmetic progression. What is $x$ ?

Problem 41.3.7

AMC 12

For how many positive integers $x$ is $\log_{10}(x - 40) + \log_{10}(60 - x) < 2$ ?

Problem 41.3.8

AMC 12

Positive real numbers $a$ and $b$ have the property that

\sqrt{\log a} + \sqrt{\log b} + \log \sqrt{a} + \log \sqrt{b} = 100

and all four terms on the left are positive integers, where $\log$ denotes the base 10 logarithm. What is $ab$ ?

Problem 41.3.10

AIME

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 41.3.11

AMC 12

Positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_2 x = \log_y 16$ and $xy = 64$ . What is $\left(\log_2 \frac{x}{y}\right)^2$ ?

Problem 41.3.12

AMC 12

What is the value of

\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right) \cdot \left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?

Problem 41.3.13

AMC 12

What is the value of

\frac{\log_2 80}{\log_{40} 2} - \frac{\log_2 160}{\log_{20} 2}?

Problem 41.3.14

AMC 12

What is the product of all the solutions to the equation

\log_{7x} 2023 \cdot \log_{289x} 2023 = \log_{2023x} 2023?

Checkpoint for section 41.2

Get 2 of 3 right to master this section. One try each, no hints.

41.3

Practice Problems

In this section you will be able to mix every log rule from this chapter on AMC 12 and AIME problems.

Time to mix it all up. These problems need the definition, the log rules, base changes and domain checks, often all at once. Every problem has three hints if you get stuck.

Some of these you have seen before: the book repeats Problem 41.3.4 as 41.3.5, Example 41.1 as 41.3.31 and Example 41.2 as 41.3.19. Your progress on them is shared.

Want every problem from this chapter on one page? Open the problem set.

Mixed practice

Problem 41.3.5

AMC 12

This problem also appears in the book as 41.3.4.

There is a unique positive integer $n$ such that

\log_2(\log_{16} n) = \log_4(\log_4 n).

What is the sum of the digits of $n$ ?

Problem 41.3.31(additional)

AMC 12

This problem also appears in the book as 41.1.

Problem 41.3.19(additional)

AIME

This problem also appears in the book as 41.2.

In a Martian civilization, all logarithms whose bases are not specified are assumed to be base $b$ , for some fixed $b \ge 2$ . A Martian student writes down

3 \log\left(\sqrt{x} \log x\right) = 56

\log_{\log x}(x) = 54

and finds that this system of equations has a single real number solution $x > 1$ . Find $b$ .

Contest cornerreal contest problems

Problem 41.3.9

AMC 12

Which of the following is the value of $\sqrt{\log_2 6 + \log_3 6}$ ?

Problem 41.3.15

AMC 12

Let $S$ be the sum of all positive real numbers $x$ for which

x^{2^{\sqrt2}} = \sqrt2^{\,2^x}.

Which of the following statements is true?

Problem 41.3.17(additional)

AIME

Real numbers $x$ and $y$ with $x, y > 1$ satisfy $\log_x\left(y^x\right) = \log_y\left(x^{4y}\right) = 10$ . What is the value of $xy$ ?

Problem 41.3.18(additional)

AMC 12

The sum of the base-10 logarithms of the divisors of $10^n$ is $792$ . What is $n$ ?

Problem 41.3.20(additional)

AIME

Let $S$ be the sum of the base 10 logarithms of all the proper divisors (all divisors of a number excluding itself) of $1000000$ . What is the integer nearest to $S$ ?

Problem 41.3.21(additional)

AIME

Positive integers $a$ and $b$ satisfy the condition

\log_2\left(\log_{2^a}\left(\log_{2^b}\left(2^{1000}\right)\right)\right) = 0.

Find the sum of all possible values of $a + b$ .

Problem 41.3.22(additional)

AIME

The sequence $a_1, a_2, \dots$ is geometric with $a_1 = a$ and common ratio $r$ , where $a$ and $r$ are positive integers. Given that $\log_8 a_1 + \log_8 a_2 + \cdots + \log_8 a_{12} = 2006$ , find the number of possible ordered pairs $(a, r)$ .

Problem 41.3.24(additional)

AIME

The increasing geometric sequence $x_0, x_1, x_2, \dots$ consists entirely of integral powers of $3$ . Given that

\sum_{n=0}^{7} \log_3(x_n) = 308 \quad\text{and}\quad 56 \le \log_3\left(\sum_{n=0}^{7} x_n\right) \le 57,

find $\log_3(x_{14})$ .

Problem 41.3.26(additional)

AMC 12

For all integers $n$ greater than $1$ , define $a_n = \frac{1}{\log_n 2002}$ . Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$ . Then $b - c$ equals

Problem 41.3.27(additional)

AMC 12

The numbers $\log\left(a^3b^7\right)$ , $\log\left(a^5b^{12}\right)$ , and $\log\left(a^8b^{15}\right)$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log b^n$ . What is $n$ ?

Problem 41.3.28(additional)

AIME

Determine the value of $ab$ if $\log_8 a + \log_4 b^2 = 5$ and $\log_8 b + \log_4 a^2 = 7$ .

Problem 41.3.29(additional)

AIME

The number

\frac{2}{\log_4 2000^6} + \frac{3}{\log_5 2000^6}

can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .

Problem 41.3.30(additional)

AMC 12

The domain of the function $f(x) = \log_{\frac12}\left(\log_4\left(\log_{\frac14}\left(\log_{16}\left(\log_{\frac{1}{16}} x\right)\right)\right)\right)$ is an interval of length $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m + n$ ?

Problem 41.3.32(additional)

AMC 12

Let $m > 1$ and $n > 1$ be integers. Suppose that the product of the solutions for $x$ of the equation

8(\log_n x)(\log_m x) - 7\log_n x - 6\log_m x - 2013 = 0

is the smallest possible integer. What is $m + n$ ?

Problem 41.3.33(additional)

AMC 12Challenge

Let $S$ be the set of ordered triples $(x, y, z)$ of real numbers for which

\log_{10}(x + y) = z \text{ and } \log_{10}\left(x^2 + y^2\right) = z + 1.

There are real numbers $a$ and $b$ such that for all ordered triples $(x, y, z)$ in $S$ we have $x^3 + y^3 = a \cdot 10^{3z} + b \cdot 10^{2z}$ . What is the value of $a + b$ ?