Equations involving logarithms and indices

The change-of-base formula, solving index equations by matching bases or taking logs, equations that reduce to quadratics, log equations, and exponential inequations.

The change-of-base formula

Change of base

$$\log_a x = \frac{\log_b x}{\log_b a}$$

Use it with $b = 10$ (or $b = e$) to evaluate any log on a calculator: $\log_3 20 = \dfrac{\log_{10} 20}{\log_{10} 3} \approx 2.727$.

Solving index equations

Example 1: matching bases●

Solve $3^{x+1} = 27^x$.

$3^{x+1} = 3^{3x}$

$x + 1 = 3x$, so $x = \tfrac12$.

Example 2: taking logs●●

Solve $2^x = 10$, correct to 3 decimal places.

$x = \log_2 10 = \dfrac{\log_{10} 10}{\log_{10} 2}$

$x \approx 3.322$

Solving 2ˣ = k

The solution is the x-coordinate where the dashed line meets the curve.

Try this

Why is there never a solution when $k \le 0$?

Equations reducible to quadratics

Example 3●●●

Solve $4^x - 3 \times 2^x - 4 = 0$.

Let $u = 2^x$. Then $u^2 - 3u - 4 = 0$.

4 to the x is (2 to the x) squared.

$(u - 4)(u + 1) = 0$, so $u = 4$ or $u = -1$.

$2^x = 4$ gives $x = 2$; $2^x = -1$ has no solution.

2 to the x is always positive.

Log equations

Example 4●●●

Solve $\log_2 x + \log_2 (x - 2) = 3$.

$\log_2 x(x - 2) = 3$, so $x(x - 2) = 8$.

$x^2 - 2x - 8 = 0$, so $x = 4$ or $x = -2$.

Check: $x = -2$ is not allowed, so $x = 4$.

log₂ x needs x > 0.

Inequations

Example 5●●●

Find the smallest whole number $n$ with $0.8^n < 0.1$.

Take logs: $n\log_{10} 0.8 < \log_{10} 0.1$.

$n > \dfrac{\log_{10} 0.1}{\log_{10} 0.8} \approx 10.3$

log 0.8 is negative, so dividing by it reverses the inequality.

The smallest whole number is $n = 11$.

Flashcards

Change of base

$\log_a x = \dfrac{\log_b x}{\log_b a}$

$a^x = k$

$x = \log_a k$

$4^x$ in terms of $2^x$

$(2^x)^2$

Always check log solutions

The argument of every log must be positive.

Practice

Find $\log_3 20$, correct to 3 decimal places.
Solve $2^x = 10$, correct to 3 decimal places.
Solve $3^{x+1} = 27^x$.
Solve $4^x - 3 \times 2^x - 4 = 0$.
The solution of $2^x > 50$ is:
Find the smallest whole number $n$ with $0.8^n < 0.1$.
Solve $\log_2 x + \log_2 (x - 2) = 3$.
Solve $5^{2x} = 3^{x+1}$, correct to 3 decimal places.
Solve $\log_x 81 = 4$.