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Chapter 10: Problem 5

Give the value of each expression. $$ \log 10^{9.6421} $$

Short Answer

Expert verified
9.6421

Step by step solution

01

Identify the logarithm properties

Recall that the logarithm of a power can be simplified using the property: \( \log_b(a^c) = c \log_b(a) \). In this exercise, we are dealing with base 10 logarithms.
02

Apply the property

Use the property from Step 1 to rewrite the given expression: \( \log(10^{9.6421}) = 9.6421 \log(10) \).
03

Simplify using the value of \( \log(10) \)

Since \( \log_{10}(10) = 1 \), we can substitute this into our expression: \( 9.6421 \log(10) = 9.6421 \cdot 1 = 9.6421 \).

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Key Concepts

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

Logarithm Simplification
Logarithms can seem tricky, but understanding their properties can simplify many problems. One key property is that the logarithm of a power can be simplified. This property is expressed mathematically as: \( \log_b(a^c) = c \log_b(a)\).

This means that the exponent
Power of Ten
The power of ten is particularly important when dealing with logarithms, especially base 10 logarithms, which are commonly used in scientific calculations.
The expression \( \log(10^{9.6421}) \) shows how powerful this concept is.

By knowing that any number raised to a power can be simplified using the power property, we find that \(10^{9.6421}\) becomes \(9.6421 \log(10)\).
Remember, these properties make simplifying expressions straightforward once you practice.
Base 10 Logarithms
Base 10 logarithms (also known as common logarithms) are the most frequently used logarithms.
When we say \( \log(10)\), we are referring to the logarithm with base 10.

A key feature of base 10 logarithms is that \( \log_{10}(10) = 1\).
This is because 10 raised to the power of 1 is 10.

Knowing this, we can simplify many expressions quickly. In our exercise above, the value of \( \log(10)\) simplifies to 1, allowing us to find that: \(9.6421 \log(10) = 9.6421 \).

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Most popular questions from this chapter

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ \ln e^{0.04 x}=\sqrt{3} $$

How much money must be deposited today to amount to \(\$ 1000\) in \(10 \mathrm{yr}\) at \(5 \%\) compounded continuously?

Solve each equation. Approximate solutions to three decimal places. $$ 4^{x-2}=5^{3 x+2} $$

Solve each equation. Give exact solutions. $$ \log (6 x+1)=\log 3 $$

How long, to the nearest hundredth of a year, would it take \(\$ 4000\) to double at \(3.25 \%\) compounded continuously?
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