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Chapter 4: Problem 89

Let \(f(x)=2^{x}, g(x)=(1 / 3)^{x}, h(x)=10^{x},\) and \(m(x)=e^{x} .\) Find the value of \(x\) in each equation. $$h(x)=1000$$

Short Answer

Expert verified
The value of x is 3.

Step by step solution

01

Set up the equation

Given the function for h(x), set up the equation with the provided value:
02

Express the equation in exponential form

Since h(x) = 10^x, replace h(x) with 1000 to form the equation: 10^x = 1000
03

Rewrite 1000 as an exponent

Write 1000 as a power of 10. Since 1000 = 10^3: 10^x = 10^3
04

Solve for x

Since the bases are the same, set the exponents equal to each other: x = 3

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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 a type of function where the variable appears in the exponent. They can be written in the form of \(f(x) = a^x\), where \(a\) is a positive real number called the base, and \(x\) is the exponent. These functions have unique properties:
  • The base \(a\) is always positive.
  • When \(a > 1\), the function grows rapidly.
  • When \(0 < a < 1\), the function decreases rapidly.
For example, in the function \(h(x) = 10^x\), 10 is the base. Understanding how to manipulate and solve exponential functions is critical for most advanced mathematics.
Let's explore further with the given example.
Solving Equations
Solving exponential equations often involves finding the value of the variable that makes the equation true. Here are key steps to solve an exponential equation:
  • Identify the exponential equation format, e.g., \(10^x = 1000\).
  • Rewrite the equation in a form that makes it easier to solve.
  • If possible, express both sides of the equation with the same base.
  • Set the exponents equal to each other and solve for the variable.
In the given problem, we set up the equation \(h(x) = 1000\), knowing that \(h(x) = 10^x\). Use these steps to find \(x\).
Any equation with a base can be simplified using properties of exponents.
Logarithms
Logarithms are closely related to exponential functions. The logarithm of a number is the exponent to which a base must be raised to produce that number. It is written as \( \text{log}_a(y) = x\) if and only if \(a^x = y\). This relationship helps solve exponential equations where the base is not easily matched.

For example, to solve \(10^x = 1000\) using logarithms:
  • Recognize that 1000 can be written as \(10^3\).
  • This form gives us \(10^x = 10^3\). Since the bases are the same, the exponents must be equal: \(x = 3\).
Logarithms simplify complex exponential equations, making it easier to find solutions.
Base Transformation
Base transformation is used when you need to convert one exponential expression to another base that is more convenient. This is particularly useful in problems where the bases of the exponential terms are different. The change of base formula for logarithms is helpful here and is given by:
\ \text{log}_a(b) = \frac{ \text{log}_c(b) }{ \text{log}_c(a) } \
For example, if you need to solve \((1/3)^{x} = 9\), you might transform the base to a common base like 3:
  • Rewrite \( (1/3)^x \) as \(3^{-x} \).
  • Write 9 as \(3^2\).
  • You now have \(3^{-x} = 3^2\). Set the exponents equal to each other: \(-x = 2\), so \(x = -2\).
Base transformation helps align equations for easier solving using exponent rules and properties.

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Most popular questions from this chapter

Solve \(2^{x-3}=4^{5 x-1}\).

Which exponential and logarithmic functions are increasing? Decreasing? Is the inverse of an increasing function increasing or decreasing? Is the inverse of a decreasing function increasing or decreasing? Explain.

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Consider the function \(y=\log \left(10^{n} \cdot x\right)\) where \(n\) is an integer. Use a graphing calculator to graph this function for several choices of \(n .\) Make a conjecture about the relationship between the graph of \(y=\log \left(10^{n} \cdot x\right)\) and the graph of \(y=\log (x) .\) Save your conjecture and attempt to prove it after you have studied the properties of logarithms, which are coming in Section \(4.3 .\) Repeat this exercise with \(y=\log \left(x^{n}\right)\) where \(n\) is an integer.
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