/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Solve for \(x.\) \(\log _{5} x... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 14

Solve for \(x.\) \(\log _{5} x=\frac{1}{2}\)

Short Answer

Expert verified
The solution to the equation \( \log_{5} x = 1/2 \) is \(x = \( \sqrt{5} \) \), which is approximately equal to \(2.23607\).

Step by step solution

01

Convert the Logarithmic Equation to an Exponential Equation

First, the logarithmic equation \(\log _{5} x=\frac{1}{2}\) should be converted into an exponential equation. According to the logarithm rules, the equation \(\log _{b} a = n \) is equivalent to \(b^n = a \). Using this rule, the equation \(\log _{5} x=\frac{1}{2}\) can be converted to \(5^{1/2} = x\).
02

Calculate the Value of x

Now from the resulting exponential equation in the first step \(5^{1/2} = x\), we can calculate the square root of 5 to find the value of \(x\). So, \(x=\sqrt{5}\) which approximately equals to \(x = 2.23607\) when rounded to five decimal places.

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Key Concepts

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

Exponential Equations
Exponential equations involve expressions where variables appear as exponents. In these equations, the same base appears with different exponents, making it crucial to understand how to manipulate and solve them. To solve a logarithmic equation by converting it to an exponential one, you need to understand the relationship between logarithms and exponents.
  • An equation of the form \(\log_b a = n\) can be converted to its exponential form \(b^n = a\).
  • This conversion is helpful because it allows one to isolate the variable more easily and solve for it directly.
  • In the context of the problem, converting \(\log_5 x = \frac{1}{2}\) to \(5^{1/2} = x\) is key to simplifying the equation.
Once converted to the exponential form, solving the equation becomes a straightforward task of evaluating the right side of the equation.
Logarithmic Rules
Logarithmic equations can be understood better through various rules that govern their operations. Knowing these rules is fundamental for transforming and solving equations.
  • One of the primary rules is the conversion rule, which states that a logarithm \(\log_b a = n\) can rearrange to become an exponential equation \(b^n = a\).
  • Other important rules include the product rule, the quotient rule, and the power rule for logarithms.
  • The base rule: changing the base of a logarithm can also help simplify an equation or solve it more easily.
These rules serve as tools to manipulate logarithmic expressions and solve equations effectively, allowing for a more systematic approach to tackling problems involving logarithms.
Square Root Calculations
Square root calculations often come into play when solving exponential equations. Understanding square roots is vital, particularly in cases where you convert the logarithmic equation into an exponential one and need to find the value of a variable.
  • The square root of a number \(a\) is a value which, when multiplied by itself, gives \(a\). So, \(\sqrt{5}\) is the number that, when squared, results in 5.
  • When evaluating square roots for non-perfect squares, approximations or decimal values are often used. For example, \(\sqrt{5} \approx 2.23607\).
  • These calculations are straightforward but require familiarity with basic arithmetic and sometimes a calculator for precision.
Understanding these simple calculations is crucial, as they are frequent in mathematical problem-solving, including solving exponential and logarithmic equations.

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Most popular questions from this chapter

Comparing Logarithmic Quantities In Exercises 83 and \(84,\) compare the logarithmic quantities. If two are equal, then explain why. $$ \log _{7} \sqrt{70}, \quad \log _{7} 35, \quad \frac{1}{2}+\log _{7} \sqrt{10} $$

Expanding a Logarithmic Expression In Exercises \(37-58,\) use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log 4 x^{2} y$$

Graphical Analysis Use a graphing utility to graph each function. Use the graph to find where the function is increasing and decreasing, and approximate any relative maximum or minimum values. (a) \(f(x)=x^{2} e^{-x} \quad\) (b) \(g(x)=x 2^{3-x}\)

Using Properties of Logarithms In Exercises \(15-20\) , use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\ln \left(5 e^{6}\right)$$

Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{4} 2+\log _{4} 32$$
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