Inverses of Logs and Exponentials

Today We Will...

Examine the inverse relationship between exponential and logarithmic functions. 

Open Ended

• What do you notice about the tables for each function?

Open Ended

• What do you notice about the points graphed for each function?

Open Ended

• How can we use our understanding of exponentials to help us graph logarithmic functions?

Inverse Relationships

​Let’s look at the inverse relationship between exponential & logarithmic functions for any base.  

​Desmos Illustration

Draw

Use the key features of f(x) = 2^x to identify the key features of f^-1(x) = log₂(x)

Draw

Use the key features of f(x) = 2^x to identify the key features of f^-1(x) = log₂(x)

Match

Match the inverse functions to each other.

• $$y=6^x$$

• $$y=\frac{1}{3^x}$$

• $$y=x^3$$

• $$y=10^x$$

• $$y=e^x$$

• $$y=\log_6x$$

• $$y=\log_{\frac{1}{3}}x$$

• $$y=x^{\frac{1}{3}}$$

• $$y=\log_{ }x$$

• $$y=\ln x$$

Logarithms are Exponents

The logarithm of a number is the exponent to which you must raise the given base to the produce the number.  

This expression is read as “log base b of y”  

Logarithms are Exponents

Consider these restrictions on the variables.

Reorder

Evaluate the following logarithmic expressions. 

Place them in order from least to greatest.

• $$\log_5\left(\frac{1}{125}\right)$$

• $$\log_44$$

• $$\log_39$$

• $$\log_216$$

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Draw

Frequently used Logarithms

Two frequently used logarithms are logarithms with base 10 and base e.

Multiple Select

Select all the logarithms the equal 3.

• $$\log_327$$

• $$\log_336$$

• $$\log_33$$

• $$\log_5125$$

• $$\log_{ }1000$$

Reorder

Mentally evaluate the following logarithmic expressions.  

Place them in order from least to greatest.
​

• $$\log\ \frac{1}{100}$$

• log 0.1

• log 1

• log 10

• log 100

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Complete Delta Math Expnentials and Logarithms