20.07 Further exponential equations (Extended)

Lesson

Worksheet

Practice

Lesson

Solving by using logarithms

If an exponential equation cannot be rearranged so that both sides have the same base, then there are two methods to solve such equations of the form $a^x=b$ax=b:

Method 1: Take the logarithm of both sides, or

Method 2: Rearrange the equation into logarithmic form and use change of base formula.

Both methods will let us rearrange the equation to get $x$x in terms of $\log a$loga and $\log b$logb. However, it is often easier to use method 1 of taking the logarithm of both sides.

Worked example

Example 1

Solve $12^x=30$12x=30 for $x$x as both an exact value and to three decimal places.

Think: This is an equation of the form $a^x=b$ax=b. So we can solve it by taking the logarithm of both sides (method 1). This might seem like a strange thing to do, but remember, as long as we do the same thing to both sides of an equation, it will continue to be true!

Do:

$12^x$12`x`	$=$=	$30$30
$\log12^x$`log`12`x`	$=$=	$\log30$`log`30	Take the logarithm of both sides
$x\log12$`xlog`12	$=$=	$\log30$`log`30	Using the identity $\log A^B=B\log A$`logAB`=`BlogA`
$x$`x`	$=$=	$\frac{\log30}{\log12}$`log`30`log`12
	$\approx$≈	$1.369$1.369	Using a calculator, and rounding to three decimal places

Reflect: When taking the log of both sides, it is easier to use logarithms of base $10$10 or $e$e, in order to evaluate the solution with a calculator.

For comparison, here is how we solve the same equation by rewriting it in logarithmic form (method 2):

$12^x$12`x`	$=$=	$30$30
$x$`x`	$=$=	$\log_{12}30$`log`1230	Using the relationship between exponential and logarithms
$x$`x`	$=$=	$\frac{\log_{10}30}{\log_{10}12}$`log`1030`log`1012	Using the change of base law
	$\approx$≈	$1.369$1.369	Rounding to three decimal places

Remember!

For exponential equations in the form $a^x=b$ax=b, there is exists a solution only if $b>0$b>0!

For example, $4^x=-1$4x=−1 and $7^{\left(x+3\right)}=-49$7(x+3)=−49 do not have any solutions, since we cannot raise the numbers $4$4 or $7$7 to any power and get a negative number.

Practice questions

Question 1

Which of the following is a good estimate for the value of $x$x, if $3^x=29$3x=29?

$-4−4<x<−3
A
$33<x<4
B
$22<x<3
C
$44<x<5
D

Question 2

Consider the equation $2^x=11$2x=11.

Rearrange the equation into the form $x=\frac{\log A}{\log B}$x=logAlogB.
Evaluate $x$x to three decimal places.

Question 3

Consider the equation $2^{9x-4}=90$29x−4=90.

Make $x$x the subject of the equation.
Evaluate $x$x to three decimal places.

Outcomes

0607E3.10A

Logarithmic function as the inverse of the exponential function y = a^x equivalent to x = log_a y. Solution to a^x = b as x = log b/log a.