Log and Exponential Form

Log and Exponential Form

Algebra 2

Take a look at the exponential functions $y=3^w$                          $F=20\cdot3^w$  
 

• What is the base?                        What is the base?

• What is the exponent?                What is the exponent?

•                                                       What is the coefficient (multiplier)?

 $F=20\cdot3^w$  

But what if  $F=100$  and you have to solve for w?

•  $100=20\cdot3^w$  

•  $5=3^w$  

• How do you solve for w?

The inverse of an exponential function is a logarithmic function

•  $x^a=b$  is the same as  $\log_xb=a$  

•  $3^w=5$ is the same as.... 

Multiple Choice

 $x^a=b$  is the same as  $\log_xb=a$  
So,  $3^w=5$  is the same as...

•  $\log_3w=5$  

•  $\log_w5=3$  

•  $\log_35=w$  

•  $\log_53=w$  

Convert from Exponential to Logarithmic

•  $1.\ \ 4^x=64$  

•  $2.\ \ \frac{1}{25}=x^{-2}$  

Convert from Exponential to Logarithmic

•  $1.\ \ 4^x=64$               Answer:  $\log_464=x$  

•  $2.\ \ \frac{1}{25}=x^{-2}$            Answer:  $\log_x\frac{1}{25}=-2$  

Multiple Choice

 Convert from exponential to logarithmic.

 $10^x=100,000$  

•  $\log_{10}100,000=x$  

•  $\log_x100,000=10$  

•  $\log_{100,000}x=10$  

•  $\log_{10}x=100,000$  

Multiple Choice

 Convert from exponential to logarithmic.

 $x^0=1$   

•  $\log_1x=0$  
  

•  $\log_01=x$  

•  $\log_10=x$  

•  $\log_x1=0$  

Multiple Choice

 Convert from exponential to logarithmic.

 $9^{0.5}=x$   

•  $\log_{0.5}x=9$  
  

•  $\log_9x=0.5$  

•  $\log_90.5=x$  

•  $\log_x9=0.5$  

Multiple Choice

 Convert from exponential to logarithmic.

 $5^x=25$   

•  $\log_{25}x=5$  
  

•  $\log_x5=25$  

•  $\log_525=x$  

•  $\log_5x=25$  

Convert from Logarithmic to Exponential

Remember:
  $x^a=b$  is the same as  $\log_xb=a$  

•  $\log_6x=2$  

•  $\ln\left(5x\right)=7$  
  

•  $\log2x=4$  

Convert from Logarithmic to Exponential

Remember:
  $x^a=b$  is the same as  $\log_xb=a$  

•  $\log_6x=2$               Answer:  $6^2=x$  

•  $\ln\left(5x\right)=7$             Answer:  $e^7=5x$  

•  $\log2x=4$               Answer:  $10^4=2x$  

Multiple Choice

Convert from Logarithmic to Exponential

 $\log100=x$  

•  $100^x=10$  

•  $10^x=100$  

•  $10^{100}=x$  

•  $100^{10}=x$  

Multiple Choice

Convert from Logarithmic to Exponential

 $\log_5625=x$  

•  $x^{625}=5$  

•  $5^x=625$  

•  $5^{625}=x$  

•  $x^5=625$  

Multiple Choice

Convert from Logarithmic to Exponential

 $\log_3\left(6x\right)=5$  

•  $6x^3=5$  

•  $3^{6x}=5$  

•  $5^{6x}=3$  

•  $3^5=6x$  

Multiple Choice

Convert from Logarithmic to Exponential

 $\ln x=5$  

•  $x^5=e$  

•  $e^x=5$  

•  $e^5=x$  

•  $e^{6x}=5$  

Find the value of x

•  $\log_4x=2$  

•  $4^2=x$  

•  $x=16$  

Multiple Choice

Find the value of x

 $\log_x8=3$  

•  $x=2$  

•  $x=8$  

•  $x=3$  

•  $x=\frac{8}{3}$  

Multiple Choice

Find the value of x

 $\log_525=x$  

•  $x=20$  

•  $x=5$  

•  $x=25$  

•  $x=2$  

Multiple Choice

Challenge Problem!
Find the value of x

 $\log_3\left(x+2\right)=4$  

•  $x=81$  

•  $x=10$  

•  $x=79$  

•  $x=83$  

Multiple Choice

Challenge Problem!
Find the value of x when  $y=2$  

 $y=\log_5\left(x-4\right)$  

•  $x=21$  

•  $x=6$  

•  $x=14$  

•  $x=29$