Change of Base Rule

 $\log_345$ 

We cannot write 3 to some exponent and get 45 so the change of base formula is appropriate.

By the change of base rule  $\log_345=\frac{\log45}{\log3}$  

 $\approx3.465$  

Some calculators have the ability to do this without using the change of base. You need to know this as a property of logarithms for upcoming work.

 $5\log_{12}51$  

 $=5\left(\frac{\log51}{\log12}\right)$  

 $\approx7.911$  

Rewrite as division  $\log_b8\div\log_b2$  

Apply the change of base rule  $\frac{\log8}{\log b}\div\frac{\log2}{\log b}$  

Division means to multiply by the reciprocal  $\frac{\log8}{\log b}\cdot\frac{\log b}{\log2}$  

Cancel the common factor  $\log b$  

 $\frac{\log8}{\log2}$  

Use the change of base rule backwards  $\log_28$  

Evaluate 3

Change of base rule:  $\frac{\log18}{\log a}\cdot\frac{\log a}{\log3}$  

Cancel common factor  $\log a$  

 $\frac{\log18}{\log3}$  

Use the change of base rule backwards

 $\log_3\left(18\right)$  

1. Evaluate logarithms change of base rule

2. Use the logarithm change of base rule