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Chapter 4: Problem 35

Simplify the expression. $$ \log 100,000,000 $$

Short Answer

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8

Step by step solution

01

Understand what base the logarithm uses

When no base is specified, the logarithm is assumed to be base 10. So, \( \log 100,000,000\) is the same as \( \log_{10} 100,000,000\).
02

Express the number in terms of powers of 10

100,000,000 can be written as \( 10^8 \).
03

Apply the power rule of logarithms

The power rule of logarithms states that \( \log_b (a^c) = c \log_b (a)\). So, \( \log_{10} (10^8) = 8 \log_{10} (10)\).
04

Simplify the logarithm

Since \( \log_{10} (10) = 1 \), the expression simplifies to \( 8 \times 1 = 8 \).

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Key Concepts

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

Base 10 Logarithms
Logarithms are a way of expressing large numbers through their powers. The base of a logarithm indicates the number which is raised to a power to achieve a certain number. When no base is mentioned, we use base 10 by default. This is known as the common logarithm.

For example, when you see \( \log 100,000,000\), it is the same as saying \( \log_{10} 100,000,000\). This means we want to find the power that 10 must be raised to in order to get 100,000,000.
Powers of 10
Working with powers of 10 can make calculations easier. In scientific notation, large numbers are often expressed as powers of 10. For instance, 100 can be written as \(10^2\), and 1,000 can be written as \(10^3\).

In our exercise, we see \(100,000,000\). We can rewrite this as a power of 10: \(10^8\). This expression tells us that 10 has been multiplied by itself 8 times to get 100,000,000.
Power Rule of Logarithms
The power rule is a valuable tool in simplifying logarithms. It states that \( \log_b (a^c) = c \log_b (a)\). This rule helps in breaking down logarithmic expressions by separating the exponent from the base.

For example, consider \(\ log_{10}(10^8)\). By using the power rule, we can rewrite this as \(8 \log_{10}(10)\). This simplification makes further calculation straightforward.
Simplifying Logarithmic Expressions
Simplifying logarithmic expressions reduces complex forms into simpler forms. The most basic simplification is to apply the rule that \( \log_b(b) = 1\). This tells us that any number raised to the power needed to get itself is 1.

In our case, when we have \( 8 \log_{10}(10)\), we can simplify \( \log_{10}(10) \) to 1. So, \( 8 \log_{10}(10)\) becomes \(8 \times 1 = 8\).

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Most popular questions from this chapter

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{5} 3 $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-6 e^{x}-16=0\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(3^{6 x+5}=5^{2 x}\)

Which functions are exponential? a. \(f(x)=\left(\frac{1}{\sqrt{3}}\right)^{x}\) b. \(f(x)=1^{x}\) c. \(f(x)=x^{\sqrt{3}}\) d. \(f(x)=(-2)^{x}\) e. \(f(x)=\pi^{x}\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(11^{1-8 x}=9^{2 x+3}\)
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