Solving Equations with Unknown Exponents

Video lesson · from 39:484 worked examples

One lesson video covers all of Exponentials & Logarithms, so it opens at 39:48 for this topic — not from the beginning.

If an unknown value is the exponent of a term (e.g. $10^x$ or $e^x$ ), we can solve for $x$ by taking the logarithm of both sides of the equation.

The same solution will be obtained using any base, but generally calculators only work in base 10 or base $e$ (natural logarithm).

Converting between the two forms: this is the single most repeated failure in logarithms — candidates cannot move between exponential form and logarithmic form, in one direction or the other. Practise both directions: $a^y = x \iff \log_a x = y$ .
Consistent lines of working: logarithms and exponentials are a place where each line must follow from the one before, and conversions are where working most often becomes inconsistent.
Bring the power down first: $\log(a^x) = x\log a$ is what makes the unknown solvable. Taking logs without applying the power law achieves nothing.
Take logs of both sides: and of the whole side, not term by term — $\log(a+b)$ is not $\log a + \log b$ .

Example 1

Solve $e^x=7$ .

x = \ln 7 \approx 1.946

Example 2

Solve the following:

a) $e^x = 10 \implies x = \ln 10$

b) $e^x = 1000 \implies x = \ln 1000 = 3\ln 10$

c) $2e^x = 0.3 \implies e^x = 0.15 \implies x = \ln 0.15$

d) $e^{x/2} = 5 \implies x = 2\ln 5 = \ln 25$

e) $e^{2x} = 18 \implies x = \frac{1}{2}\ln 18 = \ln \sqrt{18} = \ln (3\sqrt{2})$

f) $e^{-x^2} = 1 \implies -x^2 = \ln 1 \implies -x^2 = 0 \implies x = 0$

Example 3

Solve $7^y=9$ .

Take the natural logarithm of both sides:

\begin{aligned} \ln(7^y) &= \ln 9 \\ y \ln 7 &= \ln 9 \\ y &= \frac{\ln 9}{\ln 7} \approx 1.129 \end{aligned}

Example 4

Solve $5^{3x+1} = 40$ .

Take the natural logarithm of both sides:

\begin{aligned} \ln(5^{3x+1}) &= \ln 40 \\ (3x+1)\ln 5 &= \ln 40 \\ 3x+1 &= \frac{\ln 40}{\ln 5} \\ 3x &= \frac{\ln 40}{\ln 5} - 1 \\ x &= \frac{1}{3}\left( \frac{\ln 40}{\ln 5} - 1 \right) \approx 0.430 \end{aligned}