Logarithms

The definition of a logarithm, finding logs by writing numbers as powers, logs of fractions and roots, the calculator, and locating a log between two integers.

What a logarithm is

So $\log_2 8 = 3$ because $2^3 = 8$, and $\log_{10} 1000 = 3$ because $10^3 = 1000$.

Example 1●

Find $\log_3 81$, $\log_5 1$ and $\log_7 7$.

$81 = 3^4$, so $\log_3 81 = 4$.

$1 = 5^0$, so $\log_5 1 = 0$.

$7 = 7^1$, so $\log_7 7 = 1$.

Negative and fractional answers

Example 2●●

Find $\log_2 \tfrac18$, $\log_9 3$ and $\log_{10} 0.01$.

$\tfrac18 = 2^{-3}$, so $\log_2 \tfrac18 = -3$.

$3 = 9^{\frac12}$, so $\log_9 3 = \tfrac12$.

$0.01 = 10^{-2}$, so $\log_{10} 0.01 = -2$.

Example 3: solving with the definition●●

Solve $\log_x 49 = 2$ and $\log_2 x = 5$.

$x^2 = 49$ with $x > 0$, so $x = 7$.

$x = 2^5 = 32$.

The calculator and estimating

The log key gives base-$10$ logs. To place a log between integers, find powers either side: $2^4 = 16 < 20 < 32 = 2^5$, so $4 < \log_2 20 < 5$.

A log reads a power backwards

Blue: y = log₂ x is the reflection of y = 2ˣ in y = x. Move P to read log₂ of any number.

Try this

Where is $\log_2 x$ negative? Why can't you find $\log_2 0$?

Flashcards

$log_a x = y$ means

$a^y = x$

$log_a 1$

$0$

$log_a a$

$1$

$log_a frac1a$

$-1$

Domain of $log_a x$

$x > 0$

Practice

Find $\log_2 8$.
Find $\log_3 \tfrac19$.
Find $\log_5 1$.
Find $\log_4 2$.
Solve $\log_2 x = 5$.
Solve $\log_x 49 = 2$.
Find $\log_{10} 0.001$.
Between which two integers does $\log_2 20$ lie?
Use your calculator to find $\log_{10} 500$, correct to 2 decimal places.