Exponential Equations - Complex Equations

Step 1: Isolate the exponent.

In this case divide both sides of the equation by 12.

3^x = 13 Divide by 12

Step 2: Select the appropriate property to isolate the-variable.

Since the x is an exponent of base 3, take log₃ of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

${\log }_3{3}^x={\text{log}}_3 13$   Take log₃

Step 3: Apply the Property and solve for x.

Property 4 states $lo{g}_b{b}^x=x$. Thus the left-hand side becomes x.

To get a value for log₃ 13 you may need to change to log of base 10. This is covered as a separate topic.

In short, take the log of base 10 of 13 and divided by the log of base 10 of 3, the original base.

$lo{g}_{3}13=\frac{lo{g}_{10} 13}{lo{g}_{10} 3}=\frac{log 13}{log 3}$

x = log₃ 13 Apply Property

x = log₃ 13 Exact answer

$x=\frac{\text{log} 13}{\log 3}$  Change base

$x\approx 2.335$ Approximation

Step 1: Isolate the exponent.

In this case add 8 to both sides of the equation. Then divide both sides by 6.

6(2^(3x+1)) - 8 = 52 Original

6(2^(3x+1)) = 60   Add 8

2^(3x+1) = 10     Divide by 6

Step 2: Select the appropriate property to isolate the x-variable.

Since the x is an exponent of base 2, take log₂ of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

$lo{g}_2 2^{3x+1}=lo{g}_2 10$ Take log₂

Step 3: Apply the Property and solve for x.

Property 4 states $lo{g}_b{b}^x=x$. Thus the left-hand side becomes the exponent, 3x + 1. Now isolate the x.

To get a value for log₂ 10 you may need to change to log of base 10. This is covered as a separate topic.

In short take the log of base 10 of 10 and divided by the log of base 10 of 2, the original base.

$lo{g}_{2}10=\frac{lo{g}_{10} 10}{lo{g}_{10} 2}=\frac{log 10}{log 2}$

3x + 1 = log₂ 10   Apply Property

3x = log₂ 10 - 1    Subtract 1

$x=\frac{lo{g}_2 10}{3}-\frac{1}{3}$     Divide by 3

$x=\frac{lo{g}_2 10}{3}-\frac{1}{3}$    Exact answer

$x=\frac{1}{3} · \frac{\text{log}10}{\log 2}-\frac{1}{3}$ Change base

$x\approx 0.774$ Approximation

Step 1: Isolate the exponent.

In this case the exponent is isolated.

9^-3-x = 729 Original

Step 2: Select the appropriate property to isolate the x-variable.

Since the x is an exponent of base 9, take log₉ of both sides of the equation to isolate the x-variable, Property 4 - Inverse.

log₉ 9^-3-x = log₉ 729 Take log₉

Step 3: Apply the Property and solve for x.

Property 4 states $lo{g}_b{b}^x=x$. Thus the left-hand side becomes -3 - x. Now isolate the x.

To get a value for log₉ 729 you may need to change to log of base 10. This is covered as a separate topic.

In short take the log of base 10 of 729 and divided by the log of base 10 of 9, the original base.

$lo{g}_{9}729=\frac{lo{g}_{10} 729}{lo{g}_{10} 9}=\frac{log 729}{log 9}$

-3 - x = log₉ 729   Apply Property

-x = log₉ 729 + 3   Add 3

x = -(log₉ 729 + 3) Divide by -1

x = -(log₉ 729 + 3) Exact answer

$x=-\left(\frac{log 729}{\log 9}+3\right)$ Change base

x = 6   Exact value