Change of Base Rule

Change of Base Rule

Evaluate the logarithm. Round your answer to the nearest thousandth. (3 decimal places)

•  $\log_345$ 

• 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.

Multiple Choice

Evaluate the logarithm. Round your answer to the nearest thousandth. (3 decimal places)

 $\log_8200$  

•  $\log_8200=\frac{\log200}{\log8}\approx2.548$  

•  $\log_8200=\frac{\log8}{\log200}\approx.392$  

Evaluate the logarithm. Round your answer to the nearest thousandth. (3 decimal places)

•  $5\log_{12}51$  

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

•  $\approx7.911$  

Multiple Choice

Evaluate the logarithm. Round your answer to the nearest thousandth. (3 decimal places)

 $\log_4\left(\frac{1}{100}\right)$  

•  $\frac{\log\left(\frac{1}{100}\right)}{\log4}\approx3.322$  

•  $\frac{\log\left(\frac{1}{100}\right)}{\log4}\approx-3.322$  

•  $\frac{\log4}{\log\left(\frac{1}{100}\right)}\approx-.301$  

• none of these

Use the change of base rule to simplify

 $\frac{\log_b8}{\log_b2}$  

• 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

Simplify

 $\log_a18\cdot\log_3a$  

• 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

•  $\log_3\left(18\right)$  

Multiple Choice

Simplify

 $\frac{\log3}{\log_n3}$  

• 1

• log 3

•  $\log_n1$  

• log n

Multiple Choice

Simplify

 $\log_3a\cdot\log3$  

• 1

•  $\log_3\left(3a\right)$  

• log 3a

• log a

Homework in Khan Academy"

• 1. Evaluate logarithms change of base rule

  2. Use the logarithm change of base rule