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Chapter 4: Problem 2

Given \(f(x)=b^{x},\) then \(f^{-1}(x)=\) _____ for \(b>0\) and \(b \neq 1\).

Short Answer

Expert verified
The inverse function is \( f^{-1}(x) = \text{log}_b(x) \).

Step by step solution

01

Understand the Function

The given function is an exponential function of the form \[ f(x) = b^{x} \] where \( b \) is a positive constant greater than 0 and \( b eq 1 \). We need to find the inverse function, \( f^{-1}(x) \).
02

Swap x and y

To find the inverse function, first rewrite the function using y: \[ y = b^x \].Next, swap x and y to reflect the inverse function: \[ x = b^y \].
03

Solve for y

Solve the equation \( x = b^y \) for y. To do this, take the logarithm with base \( b \) of both sides of the equation. This results in: \[ \text{log}_b(x) = y \].
04

Write the Inverse Function

Replace y with \( f^{-1}(x) \) to express the inverse function: \[ f^{-1}(x) = \text{log}_b(x) \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Functions
Exponential functions are expressions where the variable appears as an exponent. The general form is written as \( f(x) = b^x \), where:
  • \(b\) is the base, a constant that is positive and not equal to 1.
  • \(x\) is the exponent or power, which can be any real number.
For instance, if \(b = 2\), the function \( f(x) = 2^x \) is doubling for each unit increase in \(x\). Exponential functions grow (or decay, if the base is between 0 and 1) rapidly compared to polynomial functions. This property makes them extremely important in fields like biology, finance, and physics. You often encounter them when dealing with compound interest, population growth, radioactive decay, and more.
Logarithms
A logarithm is the inverse of an exponential function. In simple terms, the logarithm tells us the power to which a base number must be raised to get a certain value. If you have the function \( b^y = x \), the logarithm of \(x\) with base \( b \) is written as:
  • \( \text{log}_b(x) = y \)
This means that if you know \( x \) and \( b \), the logarithm finds \( y \). Logarithms have several important properties that facilitate complex algebraic manipulations:
  • \( \text{log}_b(m * n) = \text{log}_b(m) + \text{log}_b(n) \)
  • \( \text{log}_b(m / n) = \text{log}_b(m) - \text{log}_b(n) \)
  • \( \text{log}_b(m^n) = n \times \text{log}_b(m) \)
These properties make logarithms handy for solving equations involving exponential functions.
Solving Equations
Solving equations involves finding the value(s) of the variable that make the equation true. For exponential equations (like \( x = b^y \)), using logarithms is key:
  • First, rewrite the equation if necessary so that it fits the form \( b^y = x \).
  • Next, apply the logarithm to both sides of the equation. The base of the logarithm should match the base of the exponent.
  • Simplify using the properties of logarithms to isolate the variable.
For example, given \( x = b^y \), taking the base-\( b \) logarithm of both sides gives you \( \text{log}_b(x) = y \). This step-by-step approach shows how logarithms serve as powerful tools to unravel exponentiation and solve for the variable. Using these methods, you can tackle other complicated forms like quadratic equations, trigonometric equations, etc.

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Most popular questions from this chapter

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{8} 5 $$

Two million \(E\). coli bacteria are present in a laboratory culture. An antibacterial agent is introduced and the population of bacteria \(P(t)\) decreases by half every \(6 \mathrm{hr}\). The population can be represented by \(P(t)=2,000,000\left(\frac{1}{2}\right)^{t / 6}\) a. Convert this to an exponential function using base \(e\). b. Verify that the original function and the result from part (a) yield the same result for \(P(0), P(6), P(12)\), and \(P(60) .\) (Note: There may be round- off error.)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}=-7 e^{x}\)

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{3}(7-5 z)+14=17\)

Determine if the given value of \(x\) is a solution to the logarithmic equation. \(\log _{4} x=3-\log _{4}(x-63)\) a. \(x=64\) b. \(x=-1\) c. \(x=32\)
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