/*! 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 13 Solve for \(x.\) \(\log _{4} x... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 13

Solve for \(x.\) \(\log _{4} x=3\)

Short Answer

Expert verified
So the solution to \(\log _{4} x=3\) is \(x = 64\).

Step by step solution

01

Convert the Logarithmic Equation To an Exponential Equation

The logarithm equation can be written as \(log_b a = n\), which equates to \(b^n = a\). Therefore, the equation \(\log _{4} x=3\) can be converted to an exponential equation: \(4^3 = x\).
02

Simplify the Exponential Equation

Calculate \(4^3\), which equals 64. Thus the equation becomes \(x = 64\).

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Key Concepts

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

Exponential Equations
Exponential equations are equations where the variable appears in the exponent. Understanding how to handle these equations is crucial for solving problems with logarithms because they often require conversion between exponential and logarithmic forms. In an exponential equation, we have a base raised to a power, represented as \(b^n = a\).
To solve an equation like this, we often need to identify the base, the exponent, and the result. In the example from the exercise \(b = 4\), \(n = 3\), and \(a = x\). This lets us easily compute \(4^3 = 64\).
Remember, understanding the terms is key:
  • Base (b): The number that gets multiplied.
  • Exponent (n): The number of times the base is multiplied by itself.
  • Result (a): The outcome of this multiplication.
Logarithm Properties
Logarithms are the inverse operation of exponentiation. They help us find the exponent that results in a given number when a certain base is used. Understanding this inverse relationship is fundamental to solving equations involving logarithms.
You can think of a logarithm \(\log_b(a) = n\) as answering the question, "To what power must the base \(b\) be raised, to produce \(a\)?"
This relationship allows us to switch between logarithmic and exponential equations. Using the formulaic conversion \(\log_b a = n\) can be rewritten as \[b^n = a\].
Essential properties of logarithms include:
  • Product Property: \(\log_b(MN) = \log_b M + \log_b N\)
  • Quotient Property: \(\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N\)
  • Power Property: \(\log_b(M^n) = n\cdot\log_b M\)
These properties are useful for simplifying complex logarithmic expressions.
Equation Solving
Equation solving is the process of finding the value of the variable that makes an equation true. When working with logarithmic and exponential equations, following an organized approach helps streamline finding solutions.
From the original exercise, the logarithmic equation \(\log_{4} x = 3\) is converted to the exponential form \(4^3 = x\). The solution process involves two main steps:
  • Convert the equation: Use the properties of logarithms to switch the equation into a manageable form, like transforming a log equation to an exponential one.
  • Solve for the variable: Once converted, compute the result of the exponential expression, which directly gives you the solution.

Breaking down the equation to these steps, it becomes easier to navigate complex problems. Practice regularly to familiarize yourself with these processes, and soon enough, solving logarithmic and exponential equations will become intuitive.

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Most popular questions from this chapter

Using the Change-of-Base Formula In Exercises \(11-14\) , evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{4} 8$$

Using the Change-of-Base Formula In Exercises \(11-14,\) evaluate the logarithm using the change-of-base formula. Round your result to three decimal places. $$\log _{3} 0.015$$

Condensing a Logarithmic Expression In Exercises \(67-82\) , condense the expression to the logarithm of a single quantity. $$2 \ln 8+5 \ln (z-4)$$

Condensing a Logarithmic Expression In Exercises \(67-82\) , condense the expression to the logarithm of a single quantity. $$\frac{1}{2}\left[\log _{4}(x+1)+2 \log _{4}(x-1)\right]+6 \log _{4} x$$

Compound Interest Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to calculate the balance of an investment when \(P=\$ 3000\) , \(r=6 \%,\) and \(t=10\) years, and compounding is done (a) by the day, (b) by the hour, (c) by the minute, and (d) by the second. Does increasing the number of compoundings per year result in unlimited growth of the balance? Explain.
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