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Chapter 4: Problem 41

Solve the logarithmic equation. $$\log _{x} 625=4$$

Short Answer

Expert verified
The solution to the logarithmic equation \( \log_{x} 625 = 4 \) is \( x = 5 \)

Step by step solution

01

Convert the logarithmic equation into its exponential form

From the given equation \( \log_{x} 625 = 4 \), This can be rewritten as \( x^4 = 625 \) using the property of logarithms.
02

Solve for 'x'

To solve for 'x', take the fourth root of both sides of the equation. The fourth root of 625 is 5, so \( x = 5 \). But, it's important to note here that negative numbers also can be considered as roots so \( x = -5 \) is also a solution of this equation.
03

Check the roots

Substitute \( x = 5 \) and \( x = -5 \) into the original logarithmic equation \( \log_{x} 625 = 4 \) and check if they hold true. When substituting \( x = 5 \), the equation holds true. But when substituting \( x = -5 \), the logarithm of a positive number with a negative base is undefined. So, \( x = -5 \) is not a solution.

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Key Concepts

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

Exponential Form
Let's start by converting a logarithmic equation to its exponential form. A logarithmic equation like \( \log_{b} a = c \) tells us that the base \( b \) raised to the power of \( c \) equals \( a \). So in exponential form, it's written as \( b^c = a \). This is an essential property because it simplifies solving logarithmic equations by allowing us to work with powers and roots.

For example, in the equation \( \log_{x} 625 = 4 \), converting to exponential form gives us \( x^4 = 625 \). Here, \( x \) is the base, \( 4 \) is the exponent, and \( 625 \) is the result or product when \( x \) is raised to the fourth power.
  • The exponential form makes it clearer how the base, exponent, and result are interrelated.
  • It transforms the problem from a logarithmic expression to solving for a simple exponentiation, easing the computation.
Fourth Root
Finding the fourth root is pivotal when solving equations where the variable is raised to the fourth power. The fourth root of a number \( n \) is a value \( a \) such that \( a^4 = n \). It's like asking what number multiplied by itself four times gives you \( n \).

In the context of our problem, \( x^4 = 625 \), you need to determine \( x \). The simplest way is to take the fourth root of \( 625 \).
  • Computationally, this can be done using a calculator or by recognizing that \( 625 = 5^4 \).
  • This means the fourth root of \( 625 \) is \( 5 \), making \( x = 5 \).
Some might think \( x = -5 \) is also valid, because \((-5)^4 = 625\) too. However, negative bases for logarithms lead to undefined results, as logarithms of positive numbers must have a positive base.
Logarithmic Properties
Understanding logarithmic properties is crucial for solving equations like \( \log_{x} 625 = 4 \). Key properties include the change of base formula, product property, quotient property, and power property of logarithms. Here, we leverage the basic definition and the inverse relation with exponentiation.

Some important points you might consider are:
  • Logarithms effectively "undo" exponentiation, making them powerful for isolating exponents in equations.
  • The base of a logarithm must be positive and not equal to 1.
  • For any \( \, b > 0, \, b eq 1 \), \( \log_b b = 1 \), which means the logarithm of a number to its own base is always 1.
  • The equation \( \log_{x} 625 = 4 \) uses the property that \( x^4 = 625 \).
Each logarithmic property aids in transforming and solving equations, making them essential tools in algebra and beyond.

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Most popular questions from this chapter

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$-2+2 \ln 3 x=17$$

The table shows the numbers \(N\) of college-bound seniors intending to major in engineering who took the SAT exam from 2008 through \(2013 .\) The data can be modeled by the logarithmic function $$N=-152,656+111,959.9 \ln t$$ where \(t\) represents the year, with \(t=8\) corresponding to 2008 . (Source: The College Board) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number } N \\\\\hline 2008 & 81,338 \\\2009 & 88,719 \\\2010 & 108,389 \\\2011 & 116,746 \\\2012 & 127,061 \\\2013 & 132,275 \\\\\hline\end{array}$$ (a) According to the model, in what year would 150,537 seniors intending to major in engineering take the SAT exam? (b) Use a graphing utility to graph the model with the data, and use the graph to verify your answer in part (a). (c) Do you think this is a good model for predicting future values? Explain.

Solve the logarithmic equation algebraically. Round the result to three decimal places. Verify your answer(s) using a graphing utility. $$\ln (x+1)^{2}=2$$

The table shows the yearly sales \(S\) (in millions of dollars) of Whole Foods Market for the years 2006 through 2013. (Source: Whole Foods Market) $$\begin{array}{|l|r|}\hline \text { Year } & \text { Salces } \\\\\hline 2006 & 5,607.4 \\\2007 & 6,591.8 \\\2008 & 7,953.9 \\\2009 & 8,031.6 \\\2010 & 9,005.8 \\\2011 & 10,108.0 \\\2012 & 11,699.0 \\\2013 & 12,917.0 \\\\\hline\end{array}$$ (a) Use the regression feature of a graphing utility to find an exponential model and a power model for the data and identify the coefficient of determination for each model. Let \(t\) represent the year, with \(t=6\) corresponding to 2006 (b) Use the graphing utility to graph each model with the data. (c) Use the coefficients of determination to determine which model fits the data better.

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$3 x \ln \left(\frac{1}{x}\right)-x=0$$
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