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Chapter 5: Problem 37

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log _{64} \frac{1}{4}=x$$

Short Answer

Expert verified
x = -\frac{1}{3}

Step by step solution

01

Convert the Logarithmic Equation to Exponential Form

Start by converting the logarithmic equation \( \log_{64} \frac{1}{4} = x \ \) to its exponential form. The equation \( \log_a b = c \ \) can be rewritten as \( \ a^c = b \ \). Applying this, the given equation becomes \( \ 64^x = \frac{1}{4} \ \).
02

Express Both Sides with the Same Base

Recognize that both 64 and \( \frac{1}{4} \) can be written as powers of 4. We know \( 64 = 4^3 \) and \( \frac{1}{4} = 4^{-1} \). Thus, replace 64 and \( \frac{1}{4} \) with their exponential equivalents: \( (4^3)^x = 4^{-1} \).
03

Simplify the Exponential Equation

Simplify the left-hand side of the equation using the power rule \( (a^m)^n = a^{mn} \): \( 4^{3x} = 4^{-1} \).
04

Set the Exponents Equal

Since the bases are the same, the exponents must be equal. Thus, set \( 3x = -1 \).
05

Solve for x

Solve the equation \( 3x = -1 \) by dividing both sides by 3: \( x = -\frac{1}{3} \).
06

Verify Using a Graphing Calculator

Verify the solution \( x = -\frac{1}{3} \) using a graphing calculator by plotting \( y = \log_{64}(\frac{1}{4}) \) and confirming that it equals \( -\frac{1}{3} \).

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Key Concepts

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

exponential form conversion
When dealing with logarithmic equations, one of the first steps is converting the logarithmic form to its exponential counterpart. The general formula is: if you have an equation in the form \(\log_a b = c\), you can convert it to \(a^c = b\). This is crucial because it makes the problem easier to work with, especially when you need to solve for the unknown variable. In our case, the equation \(\log_{64} \frac{1}{4} = x\) is converted to \(64^x = \frac{1}{4}\). Now we have a more manageable exponential equation.
same base method
After converting to exponential form, the next step is to express both sides of the equation with the same base. This is important because it allows us to set the exponents equal to each other. For our example, we need to rewrite 64 and \frac{1}{4} as powers of 4. We know that \(64 = 4^3\) and \frac{1}{4} = 4^{-1}\. Rewriting both sides gives us \((4^3)^x = 4^{-1}\). This simplifies the equation further, making it easier to handle.
solving exponents
Now with the equation in terms of the same base, we can simplify it using exponent rules. Specifically, we use the power rule \((a^m)^n = a^{mn}\). Applying this to our equation, we get \(4^{3x} = 4^{-1}\). With the same base on both sides, we can equate the exponents directly: \(3x = -1\). Finally, solve for x by dividing both sides by 3, yielding \x = -\frac{1}{3}\.
graphing calculator verification
It's always good practice to verify your solution. One effective method is using a graphing calculator. To check our solution \(x = -\frac{1}{3}\), we graph the function \(y = \log_{64}(\frac{1}{4})\) and see if it intersects the y-axis at -\frac{1}{3}. By plotting this on a graphing calculator, we can confirm that the value of \y\ at this point is indeed -\frac{1}{3}\, verifying our solution.

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Most popular questions from this chapter

Compound Interest. Suppose that \(\$ 82.000\) is invested at \(4 \frac{1}{2} \%\) interest, compounded quarterly. a) Find the function for the amount to which the investment grows after \(t\) years. b) Graph the function. c) Find the amount of money in the account at \(t=0,2\) \(5,\) and 10 years. d) When will the amount of money in the account reach \(\$ 100,000 ?\)

Consider quadratic functions ( \(a\) )-( h ) that follow. Without graphing them, answer the questions below. a) \(f(x)=2 x^{2}\) b) \(f(x)=-x^{2}\) c) \(f(x)=\frac{1}{4} x^{2}\) d) \(f(x)=-5 x^{2}+3\) e) \(f(x)=\frac{2}{3}(x-1)^{2}-3\) f) \(f(x)=-2(x+3)^{2}+1\) g) \(f(x)=(x-3)^{2}+1\) h) \(f(x)=-4(x+1)^{2}-3\) Which graph has vertex \((-3,1) ?\)

Centenarian Population. The centenarian population in the United States has grown over \(65 \%\) in the last 30 years. In \(1980,\) there were only \(32,194\) residents ages 100 and over. This number had grown to \(53,364\) by \(2010 .\) (Sources: Population Projections Program; U.S. Census Bureau; U.S. Department of Commerce; "What People Who Live to 100 Have in Common," by Emily Brandon, U.S. News and World Report, January \(7,2013\) ) The exponential function $$ H(t)=80,040.68(1.0481)^{t} $$ where \(t\) is the number of years after \(2015,\) can be used to project the number of centenarians. Use this function to project the centenarian population in 2020 and in 2050 (IMAGE CANT COPY)

Prove each of the following for any base a and any positive number \(x\). $$\log _{a}\left(\frac{1}{x}\right)=-\log _{a} x=\log _{1 / a} x$$

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