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Chapter 10: Problem 68

Solve equation. \(\log _{a} 1=0\)

Short Answer

Expert verified
The base \(a\) can be any positive number except 1.

Step by step solution

01

- Understand the Logarithm Property

Recall that for any base, \(a\), the logarithm of 1 is always 0. This means \( \log_{a} 1 = 0 \) holds true for any \(a\ > 0\) and \(a \eq 1.\)
02

- Apply the Logarithm Rule

Using the property that \(\log_{a} 1 = 0,\) this implies that any base \(a\) must satisfy the equation \(a^0 = 1.\)
03

- Conclusion

Since \(a^0 = 1\) holds for any \(a\ > 0\) and \(a \eq 1,\) the solution \(a\) can be any positive number except 1.

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Key Concepts

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

Logarithms
Logarithms might sound complicated at first, but they're just another way to write exponents. Think of them as the opposite of exponents. If you know an exponential equation, such as \(2^3 = 8\), the logarithm helps you find the exponent: \log_2(8) = 3\. The base of the logarithm is the number that you repeatedly multiply. In this case, it's 2.
  • Logarithms help us find how many times a number (the base) needs to be multiplied by itself to reach another number.
  • The equation \log_a(1) = 0\ shows that any base raised to the power of 0 is 1.
So, when you see \log_a(1) = 0\ in the exercise, it means any base greater than 0 and not equal to 1 will satisfy the equation.
Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. For example, \(a^x\) is an exponential function where 'a' is the base and 'x' is the exponent.
  • These functions grow rapidly and are used to model real-world scenarios like population growth, radioactive decay, and interest calculations.
  • Understanding the rules of exponents can help solve logarithmic equations.
In the context of solving \log_a(1) = 0\, recognizing that \(a^0 = 1\) for any base 'a' (except when 'a' = 1) is key.
This means exponentially, any base raised to zero is always 1.
Base of a Logarithm
The base of a logarithm is the number you're repeatedly multiplying. For example, in \log_2(8) = 3\, the base is 2.
A few things to remember about the base:
  • The base must be a positive number.
  • The base cannot be 1 (since any number raised to any power remains 1).
For the exercise's equation \( \: \log_a(1) = 0 \) to hold, 'a' can be any positive number except 1.
This is because \(a^0 = 1\) for any positive 'a', but if 'a' was 1, \log_1(1)\ would be undefined (since 1 raised to any power is always 1, making it impossible to find a unique exponent).

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Most popular questions from this chapter

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$ 3 \log _{p} x+\frac{1}{2} \log _{p} y-\frac{3}{2} \log _{p} z-3 \log _{p} a $$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$ \log _{10} 2=0.3010 \quad \text { and } \quad \log _{10} 9=0.9542 $$ Evaluate each logarithm by applying the appropriate rule or rules from this section. $$ \log _{10} \sqrt[4]{9} $$

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$ 3 \log _{a} 5-4 \log _{a} 3 $$

Use the change-of-base rule (with either common or natural logarithms) to find logarithm to four decimal places. \(\log _{6} \sqrt[3]{5}\)

The time \(t\) in years for an amount increasing at a rate of \(r\) (in decimal form) to double is given by $$t(r)=\frac{\ln 2}{\ln (1+r)}$$ This is called doubling time. Find the doubling time to the nearest tenth for an investment at each interest rate. (a) \(2 \% \text { (or } 0.02)\) (b) \(5 \% \text { (or } 0.05)\) (c) \(8 \% \text { (or } 0.08)\)
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