Solving Exponential and Logarithmic Equations

Solving exponential equations

The solution of equations of the form $a^x = b$, and equations which can be reduced to this form, including those that need prior algebraic manipulation; for example, $3^{2x} = 4$ and $25^x - 3 \times 5^x + 2 = 0$.

Example

Solve: $5^{2x} = 27$

We will solve this exactly [exactly means we will find an expression for the value of $x$ rather than calculating [using a calculator] and then rounding the answer; many log values are irrational [like surds] and so cannot be expressed precisely as a decimal, which is why we often use surds and log expressions rather than rounded numerical values].

We can take logs of both sides, but we need to decide which base is best – we could use base 5 here or [because $27 = 3^3$] we could use base 3. We will try both approaches just for completeness.

Approach 1: Take $\log_5$ of both sides:

$\log_5 5^{2x} = \log_5 27 = \log_5 3^3$

Simplifying:

$2x = 3 \log_5 3$

and so

$x = \frac{3}{2} \log_5 3$

Approach 2: Take $\log_3$ of both sides:

$\log_3 5^{2x} = \log_3 27 = \log_3 3^3$

Simplifying:

$2x \log_3 5 = 3$

And so $x = \frac{3}{2 \log_3 5}$.

You can check these two approaches give the same value using the change of base formula.