Coaching Centre
Logarithms
The definition of a logarithm, converting between index and logarithmic form, evaluating logarithms and solving simple log equations.
Definition
Logarithm
$$a^x = y \iff \log_a y = x$$$\log_a y$ asks: "what power of $a$ gives $y$?" So $\log_2 8 = 3$ because $2^3 = 8$.
The two graphs are reflections of each other in y = x, because a logarithm undoes an exponential.
Evaluate $\log_3 \tfrac{1}{9}$ and $\log_5 1$.
$3^{-2} = \tfrac{1}{9}$, so $\log_3 \tfrac{1}{9} = -2$.
$5^0 = 1$, so $\log_5 1 = 0$.
Solve $\log_x 49 = 2$.
$x^2 = 49$, and a base must be positive.
$x = 7$
$log_2 32$
$5$
$log_a 1$
$0$
$log_a a$
$1$
Evaluate $\log_3 \tfrac{1}{9}$.
$3^{-2}$.
Evaluate $\log_5 1$.
$5^0 = 1$.
Solve $\log_x 49 = 2$.
$x^2 = 49$, $x > 0$.
Evaluate $\log_2 8$.
$2^3$.
Evaluate $\log_{10} 1000$.
$10^3$.
Solve $\log_3 x = 4$.
$3^4$.
$2^5 = 32$ in log form is:
Base $2$, answer is the index $5$.
Evaluate $\log_9 3$.
$9^{\frac{1}{2}} = 3$.
New numbers every time. Keep going until you get five in a row.