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Chapter 13: Problem 51

Solve each logarithmic equation. $$\log _{144} w=\frac{1}{2}$$

Short Answer

Expert verified
To solve the logarithmic equation \(\log_{144}{w}=\frac{1}{2}\), rewrite it in exponential form as \(144^{\frac{1}{2}}=w\). Then, simplify the expression by finding the square root of 144, which is 12. Thus, the solution is \(w=12\).

Step by step solution

01

Rewrite the equation in exponential form

Given the equation \(\log_{144}{w}=\frac{1}{2}\), we can rewrite it in exponential form as: $$144^{\frac{1}{2}} = w$$
02

Simplify the expression

Now we need to find the square root of 144. The square root of a number x is a number that, when multiplied by itself, gives x. $$\sqrt{144} = 12$$ So, the equation becomes: $$12 = w$$
03

Write the final solution

The solution to the logarithmic equation is: $$w = 12$$

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Key Concepts

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

Exponential Form
Logarithmic and exponential forms are closely related. A logarithmic equation is commonly transformed into its exponential form to make calculations easier. In general, if you have an equation of the form \(\log_b(a) = c\),\ you can convert this into exponential form by understanding that \(a = b^c\). This transformation works because the logarithm \(c\) represents the exponent you'd raise \(b\) to in order to get the number \(a\).In the context of the exercise, we have the logarithmic equation \(\log_{144}{w} =\frac{1}{2}\). By transforming it into exponential form, we note that \(w = 144^{\frac{1}{2}}\). This conversion simplifies our path to finding the value of \(w\).
  • Understand that logarithms and exponents are interchangeable expressions of the same concept.
  • Remember, converting a logarithmic equation to an exponential form can make solving it more straightforward.
Square Root
The process of finding a square root is crucial to solving the given problem once it is in exponential form. A square root of a number \(x\) is a number \(y\) such that \(y \times y = x\). This concept is fundamental in algebra and essential for transforming expressions.In the solution step where \(w = 144^{\frac{1}{2}}\), expressing the exponent \(\frac{1}{2}\) means we are looking for the square root of 144. Square roots can often be simplified if the number is a perfect square. Here, \(144\) is a perfect square, and its square root is \(12\) because \(12\times12=144\).
  • Recognize when a number is a perfect square to find roots quickly.
  • Using exponent \(\frac{1}{2}\) denotes taking a square root in exponential expressions.
Solution Process
Solving logarithmic equations involves a systematic approach of conversion and simplification. By transforming the logarithmic form into an exponential equation, the problem becomes more manageable.In this exercise, we start with \(\log_{144}{w} = \frac{1}{2}\). The first step is to rewrite this as an exponential form: \(w = 144^{\frac{1}{2}}\). The next step involves simplifying \(144^{\frac{1}{2}}\) by calculating the square root of 144, which results in \(12\). The final step is to conclude the solution with \(w = 12\).
  • Convert logarithmic equations to exponential form for easier solution.
  • Simplify using fundamental arithmetic operations, like finding square roots.
  • Check if the calculated value satisfies the original equation.

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Most popular questions from this chapter

Given that \(\log 5=0.6990\) and \(\log 9=0.9542,\) use the propertics of logarithms to approximate the following $$\log \frac{1}{9}$$

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 \(1 .\) $$\log _{p} r-\log _{p} s$$

Find the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$g(x)=4 x-9$$

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve each problem. How much will Anna owe at the end of 4 yr if she borrows \(\$ 5000\) at a rate of \(7.2 \%\) compounded weekly?

Solve equation. \(\log _{2} r+\log _{2}(r+2)=3\)
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