03Ginversesexp-log

toc = Inverses =

Inverse operations are opposite operations.

For example,
 * addition and subtraction are inverse operations,
 * multiplication and division are inverse operations,
 * taking the square root and squaring are opposite operations.

For more about inverses, see 05A Inverse Relations.

When solving exponential equations earlier in the chapter we often rewrote the equations in their logarithmic form. Similarly when solving logarithmic equations we rewrote the equations in their exponential form. Exponential operations and logarithmic operations are also inverse operations.

We have seen: math . \quad\;a^x=y\;\iff\;log_ay=x \quad \{ \text{are equivalent equations}\} math

Recall from last year that the inverse of a function is obtained by interchanging the **//x//** and **//y//** elements.

Therefore, the natural exponential function, **//y = e//** x becomes **//x = e//** y. Then making **//y//** the subject, we h//ave **y = log**// e **x**, the natural logarithmic function.

Therefore, the natural exponential function and the natural logarithmic function are inverse functions.

Inverse properties
math \\ . \quad \log_a{a^x}=x \\. \\ . \quad a^{\log_ax}=x math

**Example 1**
Find the __inverse__ of the following (i) by hand and (ii) using the CAS calculator: math \\ . \quad (a) \quad y=e^{3x+1} \\. \\ . \quad (b) \quad y=\log_e(5x-3)-2 math

__Solution:__

math . \qquad y=e^{3x+1} math . .... ... Exchange x and y. math \\ . \qquad x=e^{3y+1} \;\; \iff \;\; \log_ex=3y+1 \\. \\ . \qquad \log_e(x)-1=3y \\. \\ . \qquad y=\dfrac{\log_e(x)-1}{3} math
 * (a) **

math . \qquad y=\log_e(5x-3)-2 math . ... ... Exchange x and y. math \\ . \qquad x=\log_e(5y-3)-2 \\. \\ . \qquad x+2=\log_e(5y-3) \;\; \iff \;\; e^{x+2}=5y-3 math . math \\ . \qquad e^{x+2}+3=5y \\. \\ . \qquad y=\dfrac{e^{x+2}+3}{5} math
 * (b) **

(ii) When using the CAS calculator to find the inverse, enter the inverse equation by interchanging **//x//** and **//y//** and then solve for **//y//**.

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