Logarithm Functions

Main idea

A logarithm answers "what power?": $\log_b x$ is the exponent you put on $b$ to get $x$ . So logs undo exponentials, and every log rule is an exponent rule read backwards.

{\htmlData{sym=1}{y}} = \log_{\htmlData{sym=0}{b}} {\htmlData{sym=2}{x}} \iff {\htmlData{sym=0}{b}}^{\htmlData{sym=1}{y}} = {\htmlData{sym=2}{x}}

What the symbols mean

$b$
Base
The number being raised to a power.
$y$
Exponent
The power b needs to make x.
$x$
Argument
The result you're trying to reach; must be positive.

\log_b(xy) = \log_b x + \log_b y, \qquad \log_b\frac xy = \log_b x - \log_b y, \qquad \log_b x^{\htmlData{sym=0}{r}} = {\htmlData{sym=0}{r}}\log_b x

What the symbols mean

$r$
Power
Comes down in front as a multiplier.

Change of base

\log_b x = \frac{{\htmlData{sym=0}{\ln}} x}{{\htmlData{sym=0}{\ln}} b}

What the symbols mean

$\ln$
Natural logarithm
The logarithm with base e.

Explanation

A logarithm is simply the answer to "what power do I need?" If $2^3 = 8$ , then the log base 2 of 8 is 3, because 3 is the power that turns 2 into 8. The definition up top has three pieces: $b$ (the base you're raising), $x$ (the number you're trying to reach) and $y$ (the power that gets you there). The rules below look like a lot, but each is an exponent rule you already know, turned around. Logs are good for undoing exponentials and for squashing huge ranges of numbers onto a manageable scale.

Evaluating. Just ask the question. $\log_2 8 = 3$ because $2^3 = 8$ . $\log_3\frac19 = -2$ because $3^{-2} = \frac19$ ; a negative log means the number is between 0 and 1.

Special logarithms. Two bases get their own names: $\ln x = \log_e x$ (the natural log) and $\log x = \log_{10} x$ (the common log).

Facts that follow from the definition. $\log_b 1 = 0$ because $b^0 = 1$ , and $\log_b b = 1$ because $b^1 = b$ . Also $\log_b b^x = x$ and $b^{\log_b x} = x$ : doing an exponential and then its log (or the other way round) gets you back where you started.

The rules mirror exponents. Since a log is an exponent, and multiplying powers adds their exponents, the log of a product has to be a sum of logs. Dividing subtracts exponents, so logs of quotients subtract. And raising to a power multiplies exponents, so the power comes down in front.

Domain. $\log_b x$ only takes positive inputs, because $b$ to any power is positive; there's no power of 2 that gives $-4$ or $0$ . Its graph is the graph of $b^x$ reflected across $y = x$ : it passes through $(1, 0)$ , rises slowly, and has a vertical asymptote at $x = 0$ .

A common mistake. Because $\log(xy) = \log x + \log y$ , it's tempting to split $\log(x + y)$ into $\log x + \log y$ too. There's no such rule. Try $\log_{10}(10 + 10) = \log_{10} 20 \approx 1.3$ , while $\log_{10}10 + \log_{10}10 = 2$ . Logs turn products into sums, not sums into anything simpler.

See it

The log is the exponential read backwards. Every point (a, 2ᵃ) on y = 2ˣ has a twin (2ᵃ, a) on y = log₂ x, with the coordinates swapped, so the two graphs are mirror images across y = x. That's why log₂ 8 = 3, why logs of numbers between 0 and 1 come out negative, and why log₂ of 0 or a negative number doesn't exist.

Worked examples

Example 1

Write $\ln\dfrac{x^2\sqrt y}{z^3}$ in terms of $\ln x$ , $\ln y$ , $\ln z$ .

1
Split the quotient into a difference: $\ln(x^2\sqrt y) - \ln z^3$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Evaluate $\log_2 32$ .

Problem 2+5 XP

Evaluate $\log_3\frac19$ .

Problem 3+5 XP

Evaluate $\ln e^5$ .

Problem 4+5 XP

Combine $2\ln x + \ln 3$ into a single logarithm.

Problem 5+5 XP

Use change of base to evaluate $\log_5 17$ (to three decimals).

Problem 6+5 XP

Acidity is measured by $\text{pH} = -\log_{10}[\text{H}^+]$ . Seawater has $[\text{H}^+] = 3.16 \times 10^{-9}$ mol/L. Find its pH.

Problem 7+5 XP

Sound level is $L = 10\log_{10}\frac{I}{I_0}$ decibels, where $I_0$ is the quietest audible intensity. How loud is a sound 1000 times as intense as $I_0$ ?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

One speaker plays music at 70 dB. How many identical speakers playing together would it take to reach 90 dB?

Where it's used

Decibels, the Richter scale and pH are logarithmic scales; computer scientists use $\log_2 n$ for the number of steps in binary search.

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Exponential Functions Solving Exponential Equations

Logarithm Functions

Main idea

A logarithm answers "what power?": $\log_b x$ is the exponent you put on $b$ to get $x$ . So logs undo exponentials, and every log rule is an exponent rule read backwards.

{\htmlData{sym=1}{y}} = \log_{\htmlData{sym=0}{b}} {\htmlData{sym=2}{x}} \iff {\htmlData{sym=0}{b}}^{\htmlData{sym=1}{y}} = {\htmlData{sym=2}{x}}

What the symbols mean

$b$
Base
The number being raised to a power.
$y$
Exponent
The power b needs to make x.
$x$
Argument
The result you're trying to reach; must be positive.

\log_b(xy) = \log_b x + \log_b y, \qquad \log_b\frac xy = \log_b x - \log_b y, \qquad \log_b x^{\htmlData{sym=0}{r}} = {\htmlData{sym=0}{r}}\log_b x

What the symbols mean

$r$
Power
Comes down in front as a multiplier.

Change of base

\log_b x = \frac{{\htmlData{sym=0}{\ln}} x}{{\htmlData{sym=0}{\ln}} b}

What the symbols mean

$\ln$
Natural logarithm
The logarithm with base e.

Explanation

Evaluating. Just ask the question. $\log_2 8 = 3$ because $2^3 = 8$ . $\log_3\frac19 = -2$ because $3^{-2} = \frac19$ ; a negative log means the number is between 0 and 1.

Special logarithms. Two bases get their own names: $\ln x = \log_e x$ (the natural log) and $\log x = \log_{10} x$ (the common log).

See it

Worked examples

Example 1

Write $\ln\dfrac{x^2\sqrt y}{z^3}$ in terms of $\ln x$ , $\ln y$ , $\ln z$ .

1
Split the quotient into a difference: $\ln(x^2\sqrt y) - \ln z^3$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Evaluate $\log_2 32$ .

Problem 2+5 XP

Evaluate $\log_3\frac19$ .

Problem 3+5 XP

Evaluate $\ln e^5$ .

Problem 4+5 XP

Combine $2\ln x + \ln 3$ into a single logarithm.

Problem 5+5 XP

Use change of base to evaluate $\log_5 17$ (to three decimals).

Problem 6+5 XP

Acidity is measured by $\text{pH} = -\log_{10}[\text{H}^+]$ . Seawater has $[\text{H}^+] = 3.16 \times 10^{-9}$ mol/L. Find its pH.

Problem 7+5 XP

Sound level is $L = 10\log_{10}\frac{I}{I_0}$ decibels, where $I_0$ is the quietest audible intensity. How loud is a sound 1000 times as intense as $I_0$ ?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

One speaker plays music at 70 dB. How many identical speakers playing together would it take to reach 90 dB?