Logarithm laws and exponential equations

The product, quotient and power laws for logarithms, simplifying log expressions, and solving exponential equations using logarithms.

Laws

Example 1●●

Simplify $2\log_{10} 5 + \log_{10} 4$.

$\log_{10} 25 + \log_{10} 4 = \log_{10} 100$

$= 2$

Example 2●●

Solve $2^x = 10$ to 2 d.p.

Take $\log_{10}$ of both sides: $x\log 2 = \log 10$.

$x = \dfrac{\log 10}{\log 2} \approx 3.32$

Flashcards

$log 2 + log 5$ (base 10)

$1$

$log_a x^n$

$n\log_a x$

Solve $a^x = b$

$x = \dfrac{\log b}{\log a}$

Practice

Simplify $2\log_{10} 5 + \log_{10} 4$.
Solve $2^x = 10$ (2 d.p.).
Simplify $\log_{10} 2 + \log_{10} 5$.
Simplify $\log_3 54 - \log_3 2$.
Solve $3^x = 50$ (2 d.p.).
A $\$1000$ investment grows by $5\%$ p.a. After how many whole years does it first double? ($1.05^n \ge 2$)
Simplify $\log_2 48 - \log_2 3$.
Simplify $\log_2 8^2$.