Problem 33
Question
Evaluate each expression without using a calculator. $$\log _{64} 8$$
Step-by-Step Solution
Verified Answer
The value of \( \log_{64} 8 \) is 0.5
1Step 1 : Understand the logarithm equation
Express the logarithm equation \( \log_{64} 8 \) as the exponential equation \( 64^x = 8 \)
2Step 2 : Express the numbers with the same base
Most notably, it is possible to express both 64 and 8 as powers of 2: \( 64 = 2^6 \) and \( 8 = 2^3 \). This allows us to represent the exponential equation from Step 1 with a common base: \( (2^6)^x = 2^3 \)
3Step 3 : Simplify the exponential equation
Simplifying the left side yields: \( 2^{6x} = 2^3 \). We can reason that if \( 2^{6x} = 2^3 \), then the exponents must be equal, so \( 6x = 3 \)
4Step 4 : Solve for x
Solving for x, we get \( x = \frac{3}{6} = 0.5 \)
Other exercises in this chapter
Problem 32
In Exercises \(25-34,\) begin by graphing \(f(x)-2^{x}\). Then use transformations of this graph to graph the given function. Be sure to graph and give equation
View solution Problem 33
Use the exponential growth model, \(A=A_{0} e^{t_{i}},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(2 A_{0}\) ) is given
View solution Problem 33
In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions
View solution Problem 33
In Exercises \(25-34,\) begin by graphing \(f(x)-2^{x}\). Then use transformations of this graph to graph the given function. Be sure to graph and give equation
View solution