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Chapter 13: Problem 32

Solve each exponential equation. $$4^{y}=16$$

Short Answer

Expert verified
The solution to the given exponential equation \(4^y = 16\) is \(y=2\).

Step by step solution

01

Rewrite the equation with the same base

We need to rewrite 16 as a power of 4. We know that 4^2=16, so the given equation becomes: $$4^y=4^2$$
02

Use the property of exponents a^x = a^y 鈬� x = y

Since both sides of the equation have the same base (4), we can equate the exponents y and 2: $$y=2$$
03

Write the solution

The solution to the given exponential equation 4^y = 16 is: $$y=2$$

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Key Concepts

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

Exponents
Exponents, often seen in equations as small numbers or symbols next to a base number, represent repeated multiplication. For example, in the expression \(4^2\), the base is 4, and the exponent is 2. This tells us to multiply 4 by itself, which gives us 16. Understanding this concept is crucial when dealing with exponential equations as it forms the basis for many algebraic manipulations.
Recognizing common powers can often simplify problems. For example, knowing that \(4^2 = 16\) helps us rewrite numbers in terms of powers of other numbers. This rewriting is useful in solving exponential equations, a key skill in algebra.
Equation Solving
Solving an equation means finding the value of the variable that makes the equation true. In the case of exponential equations, we often seek to express both sides with the same base to simplify the comparison.
Consider the equation \(4^y = 16\). By expressing 16 as a power of 4, or \(4^2\), both sides have the same base. This equivalence simplifies the equation solving process immensely, as similar bases allow us to compare exponents directly. Rewriting as \(4^y = 4^2\), we can see that \(y\) must equal 2.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging or modifying an equation in order to solve it, often by using properties of operations and exponents. When solving exponential equations, we frequently use these manipulations to handle exponents and variables.
Here, algebraic manipulation began with recognizing that both sides of the equation \(4^y = 16\) could be expressed in terms of the same base. By rewriting 16 as \(4^2\), we simplified the equation by using the property \(a^x = a^y\) to conclude \(x = y\). Such strategic manipulations are at the heart of problem-solving in algebra, enabling us to find solutions efficiently.

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Most popular questions from this chapter

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$\log _{p} r-\log _{p} s$$

Solve equation. \(\log _{3} y+\log _{3}(y-8)=2\)

Find the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$f(x)=-2 x+5$$

Determine whether each function is one-to-one. If it is one-to-one, find its inverse. $$h=\\{(-5,-16),(-1,-4),(3,8)\\}$$

Find the inverse of each one-to-one function. $$g(x)=\sqrt{x+3}, x \geq-3$$
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