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Chapter 5: Problem 89

Use the change-of-base rule to find an approximation for each logarithm. $$\log _{5} 10$$

Short Answer

Expert verified
\(\log_5 10 \approx 1.43067\).

Step by step solution

01

Understanding the Change-of-Base Formula

The change-of-base formula allows us to evaluate logarithms using a different base, usually 10 or e, which are easier to compute with calculators. The formula is: \[\log_b a = \frac{\log_c a}{\log_c b}\]where \(c\) is the new base. Common choices for \(c\) are 10 (common logarithm) or \(e\) (natural logarithm).
02

Apply the Change-of-Base Formula

We need to find \(\log_5 10\). Using the change-of-base formula with base 10, we write:\[\log_5 10 = \frac{\log_{10} 10}{\log_{10} 5}\]
03

Calculate Each Logarithm Using a Calculator

Using a calculator, we find: \(\log_{10} 10 = 1\) because 10 raised to the power of 1 is 10, and \(\log_{10} 5 \approx 0.69897\).
04

Compute the Approximation

Now, substitute the values into the change-of-base formula:\[\log_5 10 = \frac{1}{0.69897} \approx 1.43067\] So, \(\log_5 10 \approx 1.43067\).

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Key Concepts

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

Logarithms
Logarithms are a mathematical operation that help us deal with exponential relationships. They are essentially the inverse of exponentiation. This means if you know a result in an exponential equation and you want to find one of the exponents, you would use logarithms. For example, in the equation \( 10^x = 100 \), we can find \( x \) by using logarithms.We have different bases for logarithms:
  • Logarithm base 10: This is known as the common logarithm. Often written as \( \log \).
  • Logarithm base \( e \): This is known as the natural logarithm. Represented as \( \ln \).
  • Logarithm base 2: Used in computing, known as binary logarithm.
By switching bases using a formula known as the change-of-base formula, we can calculate logarithms more easily, particularly when using calculators that have built-in functions for specific bases.
Common Logarithm
The common logarithm is simply a logarithm with a base of 10. It's widely used in science and engineering because it simplifies expressions with powers of 10. When you see \( \log \) without a specified base, it's usually assumed to be base 10.One main property of the common logarithm is:
  • \( \log_{10} 10 = 1 \): because 10 to the power of 1 is 10.
You will often use common logarithms in your calculations involving the change-of-base formula. This is because most scientific calculators have a dedicated \( \log \) button that automatically uses base 10. As a result, using common logarithms makes computations straightforward when evaluating logarithms with different bases.
Natural Logarithm
Natural logarithms use the base \( e \), where \( e \approx 2.718 \). This base, \( e \), arises naturally in the context of continuous growth or decay, making natural logarithms very important in calculus and complex mathematics.Whenever you see \( \ln \), it denotes a logarithm with base \( e \). Some key characteristics include:
  • The natural logarithm of \( e \), \( \ln e = 1 \), because \( e^1 = e \).
  • They are useful in solving differential equations and situations involving compound interest.
Natural logarithms provide a framework for problems involving rates of exponential growth, which appear in fields ranging from natural sciences to economics. Understanding the natural logarithm allows you to connect exponential functions to their inverses, simplifying complex calculations.

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Most popular questions from this chapter

The interest rate stated by a financial institution is sometimes called the nominal rate. If interest is compounded, the actual rate is, in general, higher than the nominal rate, and is called the effective rate. If \(r\) is the nominal rate and \(n\) is the number of times interest is compounded annually, then $$R=\left(1+\frac{r}{n}\right)^{n}-1$$ is the effective rate. Here, \(R\) represents the annual rate that the investment would earn if simple interest were paid. Estimate the effective rate if the nominal rate is \(4.5 \%\) and interest is compounded daily \((n=365)\)

In general, it is not possible to find exact solutions analytically for equations that involve exponential or logarithmic functions together with polynomial, radical, and rational functions. Solve each equation= using a graphical method, and express solutions to the nearest thousandth if an approximation is appropriate. $$\log x=x^{2}-8 x+14$$

Emissions Governments could reduce carbon emissions by placing a tax on fossil fuels. The costbenefit equation $$\ln (1-P)=-0.0034-0.0053 x$$ estimates the relationship between a tax of \(x\) dollars per ton of carbon and the percent \(P\) reduction in emissions of carbon, where \(P\) is in decimal form. Determine \(P\) when \(x=60 .\) Interpret the result. (Source: Clime, W., The Economics of Global Warming, Institute for International Economics.)

Suppose that \(\$ 2500\) is invested in an account that pays interest compounded continuously. Find the amount of time that it would take for the account to grow to the given amount at the given rate of interest. Round to the nearest tenth of a year. \(\$ 3000\) at \(3.75 \%\)

Assume that \(f(x)=a^{x},\) where \(a>1\) If \(a=e,\) what is an equation for \(y=f^{-1}(x) ?\) (You need not solve for \(y .\) )
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