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Chapter 6: Problem 107

Solve. $$5^{x}=625$$

Short Answer

Expert verified
x = 4

Step by step solution

01

Identify the Base

Notice that the equation is in the form of an exponential equation with base 5: \[5^x = 625\]
02

Express 625 as a Power of 5

Find a power of 5 that equals 625. Recognize that: \[625 = 5^4\]
03

Rewrite the Equation with the Same Base

Rewrite the given equation with the same base on both sides: \[5^x = 5^4\]
04

Set Exponents Equal

Since the bases are the same, set the exponents equal to each other: \[x = 4\]
05

Solve for x

The solution to the equation is simple and direct: \[x = 4\]

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Key Concepts

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

Solving Exponential Equations
When working with exponential equations, the goal is to find the value of the variable that makes the equation true. In this particular problem, we need to solve for \(x\) in the equation \(5^x = 625\).
To achieve this, we can follow a series of steps:
  1. Identify the base of the exponential expression.
  2. Rewrite both sides of the equation with the same base if necessary.
  3. Set the exponents equal to each other once the bases are the same.
  4. Solve for the variable.

By following these steps, the problem becomes simpler, and we can isolate the variable to find the solution.
Exponent Properties
Understanding the properties of exponents is crucial when solving exponential equations. Here are some fundamental properties:
  • \(a^m \times a^n = a^{m+n}\)
  • \(\frac{a^m}{a^n} = a^{m-n}\)
  • \((a^m)^n = a^{mn}\)
In the provided exercise, we specifically used the property that if the bases are identical, the exponents can be equated:
\(a^m = a^n \rightarrow m = n\).
This property allows us to simplify equations and directly solve for the unknown variable once the bases match.
Base Identification
Identifying the base of an exponential expression is the first crucial step. In the equation \(5^x = 625\), the base is clearly 5.

To make the equation easier to solve, we look to express 625 as a power of 5. Recognizing that:
\(625 = 5^4\)
allows us to rewrite the original equation as \(5^x = 5^4\).
Once the bases on each side of the equation are the same, solving for \(x\) becomes straightforward because we can set the exponents equal:
\(x = 4\).
This identification process simplifies the solving of exponential equations significantly.

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Most popular questions from this chapter

Given the function value and the quadrant restriction, find \(\theta\). FUNCTION VALUE = \(\csc \theta=1.0480\) INTERVAL = \(\left(0^{\circ}, 90^{\circ}\right)\) \(\boldsymbol{\theta}\) = ____

Solve. $$\log _{7} x=3$$

Given that \(\cos \theta=0.9651,\) find \(\csc \left(90^{\circ}-\theta\right)\).

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Make a hand-drawn graph of the function. Then check your work using a graphing calculator. $$f(x)=e^{x / 2}$$
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