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Inverse Properties of Logarithms

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Inverse Properties of Logarithmic Functions

If you continue to study mathematics into college, you may take a course called Differential Equations. There you will learn that the solution to the differential equation y' = y is the general function y = Ce^x . What is the inverse of this function?

Guidance

By the definition of a logarithm, it is the inverse of an exponent. Therefore, a logarithmic function is the inverse of an exponential function. Recall what it means to be an inverse of a function. When two inverses are composed (see the Inverse Functions concept), they equal x . Therefore, if f(x)=b^x and g(x)=\log_b x , then:

f \circ g=b^{\log_b x}=x and g \circ f =\log_b b^x=x

These are called the Inverse Properties of Logarithms.

Example A

Find:

a) 10^{\log 56}

b) e^{\ln6} \cdot e^{\ln2}

Solution: For each of these examples, we will use the Inverse Properties.

a) Using the first property, we see that the bases cancel each other out. 10^{\log 56}=56

b) Here, e and the natural log cancel out and we are left with 6 \cdot 2=12 .

Example B

Find \log_4 16^x

Solution: We will use the second property here. Also, rewrite 16 as 4^2 .

\log_4 16^x=\log_4 (4^2)^x=\log_4 4^{2x}=2x

Example C

Find the inverse of f(x)=2e^{x-1} .

Solution: See the Finding the Inverse concept for the steps on how to find the inverse.

Change f(x) to y . Then, switch x and y .

& y=2e^{x-1} \\& x=2e^{y-1}

Now, we need to isolate the exponent and take the logarithm of both sides. First divide by 2.

& \frac{x}{2}=e^{y-1} \\& \ln \left(\frac{x}{2}\right)= \ln e^{y-1}

Recall the Inverse Properties from earlier in this concept. \log_b b^x=x ; applying this to the right side of our equation, we have \ln e^{y-1}=y-1 . Solve for y .

& \ln \left(\frac{x}{2}\right)=y-1 \\& \ln \left(\frac{x}{2}\right)+1=y

Therefore, \ln \left(\frac{x}{2}\right)+1 is the inverse of 2e^{y-1} .

Intro Problem Revisit Switch x and y in the function y = Ce^x and then solve for y .

x = Ce^y\\\frac{x}{C} = e^y\\ln \frac{x}{C} = ln (e^y)\\ln \frac{x}{C} = y

Therefore, the inverse of y = Ce^x is y = ln \frac{x}{C} .

Guided Practice

1. Simplify 5^{\log_5 6x} .

2. Simplify \log_9 81^{x+2} .

3. Find the inverse of f(x)=4^{x+2}-5 .

Answers

1. Using the first inverse property, the log and the base cancel out, leaving 6x as the answer.

5^{\log_5 6x}=6x

2. Using the second inverse property and changing 81 into 9^2 we have:

\log_9 81^{x+2} &= \log_9 9^{2(x+2)} \\&= 2(x+2) \\&= 2x+4

3. Follow the steps from Example C to find the inverse.

f(x) &= 4^{x+2}-5 \\y &= 4^{x+2}-5 \\x &= 4^{y+2}-5 \\x+5 &= 4^{y+2} \\\log_4 (x+5) &= y+2 \\\log_4 (x+5)-2 &= y

Vocabulary

Inverse Properties of Logarithms
\log_b b^x=x and b^{\log_b x}=x, b \ne 1

Practice

Use the Inverse Properties of Logarithms to simplify the following expressions.

  1. \log_3 27^x
  2. \log_5 \left(\frac{1}{5}\right)^x
  3. \log_2 \left(\frac{1}{32}\right)^x
  4. 10^{\log(x+3)}
  5. \log_6 36^{(x-1)}
  6. 9^{\log_9(3x)}
  7. e^{\ln(x-7)}
  8. \log \left(\frac{1}{100}\right)^{3x}
  9. \ln e^{(5x-3)}

Find the inverse of each of the following exponential functions.

  1. y=3e^{x+2}
  2. f(x)=\frac{1}{5}e^\frac{x}{7}
  3. y=2+e^{2x-3}
  4. f(x)=7^{\frac{3}{x}+1-5}
  5. y=2(6)^\frac{x-5}{2}
  6. f(x)=\frac{1}{3}(8)^{\frac{x}{2}-5}

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