Problem 91
Question
Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\log _{4} 16^{3.4}$$.
Step-by-Step Solution
Verified Answer
The exact value of the logarithm \( \log _{4} 16^{3.4} \) is 6.8.
1Step 1: Apply the Power Rule
By using the power rule of logarithms which allows us to move the exponent in \(16^{3.4}\) out in front of the log, this expression can be rewritten as \(3.4 \log _{4} 16\).
2Step 2: Simplify the logarithm
Now, we need to simplify log base 4 of 16. In other words, we are looking for the exponent that would need to be applied to 4 to equal 16. Since \( 4^2 = 16 \), this means that \( \log _{4} 16 = 2 \). Substitute this back into our equation to get \( 3.4 * 2 \).
3Step 3: Compute the final result
Multiplying \( 3.4 \) by \( 2 \) yields \( 6.8 \). This is the exact value of the logarithm.
Key Concepts
Power Rule of LogarithmsLogarithm SimplificationExponents
Power Rule of Logarithms
The power rule is a fundamental property of logarithms. It's particularly useful when dealing with an exponent inside the argument of a logarithm. This rule states that the logarithm of a power can be simplified by bringing the exponent in front of the logarithm. In mathematical terms: \( \log_b(a^n) = n \cdot \log_b(a) \).
For instance, in the exercise given, we see \( \log_{4} 16^{3.4} \). Using the power rule, we can simplify this to \( 3.4 \cdot \log_{4} 16 \). This transformation is valuable as it converts an exponentiation problem into a multiplication one, making it simpler to solve.
For instance, in the exercise given, we see \( \log_{4} 16^{3.4} \). Using the power rule, we can simplify this to \( 3.4 \cdot \log_{4} 16 \). This transformation is valuable as it converts an exponentiation problem into a multiplication one, making it simpler to solve.
Logarithm Simplification
Logarithm simplification involves breaking down a logarithmic expression into a more manageable form. In many cases, this means finding the simplest form of a log expression by interpreting what the base and the result signify.
In our exercise, you encounter the expression \( \log_{4} 16 \). Here, you should find the exponent to which the base 4 must be raised to yield 16. By calculating, you can determine that \( 4^2 = 16 \), leading to \( \log_{4} 16 = 2 \). This value then replaces \( \log_{4} 16 \) in our equation, easing the computation process by reducing the complexity into a straightforward multiplication.
In our exercise, you encounter the expression \( \log_{4} 16 \). Here, you should find the exponent to which the base 4 must be raised to yield 16. By calculating, you can determine that \( 4^2 = 16 \), leading to \( \log_{4} 16 = 2 \). This value then replaces \( \log_{4} 16 \) in our equation, easing the computation process by reducing the complexity into a straightforward multiplication.
Exponents
Exponents represent repeated multiplication and are seen in expressions like \( a^n \), where \( a \) is the base, and \( n \) is the exponent. It means you multiply \( a \) by itself \( n \) times. Understanding how exponents relate to logarithms is essential, especially when simplifying logarithmic expressions involving powers.
For example, in the expression \( 16^{3.4} \), the base is 16, and 3.4 is the exponent. In problems involving logarithms, identifying an exponent that simplifies the equation can greatly assist in solving logarithmic expressions, as demonstrated by recognizing that \( 4^2 = 16 \), which facilitated our simplification of \( \log_{4} 16 \) in the step-by-step solution.
For example, in the expression \( 16^{3.4} \), the base is 16, and 3.4 is the exponent. In problems involving logarithms, identifying an exponent that simplifies the equation can greatly assist in solving logarithmic expressions, as demonstrated by recognizing that \( 4^2 = 16 \), which facilitated our simplification of \( \log_{4} 16 \) in the step-by-step solution.
Other exercises in this chapter
Problem 91
Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$g(t)=e^{0.09 t}-3$$
View solution Problem 91
Find the domain, vertical asymptote, and \(x\) -intercept of the logarithmic function, and sketch its graph by hand. Verify using a graphing utility. $$g(x)=\ln
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Think About It In Exercises \(89-92,\) place the correct symbol \(( )\) between the two numbers. $$5^{-3} \quad 3^{-5}$$
View solution Problem 92
Use a graphing utility to graph the function and approximate its zero accurate to three decimal places. $$h(t)=e^{-0.125 t}-8$$
View solution