The graphs of the functions y=a^x & y=logaX are symmetric with respect to the line y=???

Question

very confused please provide answer. the a is underscored under log 

Answer 1

The line of symmetry would be y=x.

 

The exponential function and logarithmic functions are inverses of each other. For all functions that are inverses of each other, if you switch x and y, and then solve for y, you will get the other function.

 

Let's look at this with some real numbers.

 

100 = 10²

This is written in exponential form.

 

Written in logarithmic form, it looks like this:

2 = log₁₀ 100

 

 

 

Now, let's look at the function y=10^x

if x=0, y(0) = 10⁰ = 1

if x=1, y(1) = 10¹ = 10

if x=2, y(2) = 10² = 100

 

Now, let's calculate the inverse of that function:

x=10^y

log₁₀ x = log₁₀ 10^y

log₁₀ x = y

If we were to substitute the y-values above in place of x into this new function, watch what happens:

if x=1, y(1) = log₁₀ (1) = 0

if x=10, y(10) = log₁₀ (10) = 1

if x=100, y(100) = log₁₀ (100) = 2

 

Notice when you take the inverse, the x- and y-values switch. If you were to plot these on a graph, that means that graphs are symmetric over the line y=x