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Chapter 18: Problem 2

Solve for \(x\) to three significant digits. $$(7.26)^{x}=86.8$$

Short Answer

Expert verified
The value of \(x\) to three significant digits is approximately 2.25.

Step by step solution

01

Take the logarithm of both sides

To solve for the exponent variable, take the natural logarithm (ln) of both sides of the equation to utilize the property of logarithms that allows the exponent to be brought down as a coefficient. The equation becomes: \( \ln((7.26)^x) = \ln(86.8) \).
02

Apply the power rule of logarithms

Use the power rule, which states that \( \ln(a^b) = b \ln(a) \), to bring down the exponent. The equation simplifies to: \( x \cdot \ln(7.26) = \ln(86.8) \).
03

Divide both sides by \(\ln(7.26)\)

To isolate the variable \(x\), divide both sides of the equation by \(\ln(7.26)\). The equation becomes: \( x = \frac{\ln(86.8)}{\ln(7.26)} \).
04

Calculate the value of \(x\)

Now, calculate both logarithms and divide them to find the value of \(x\). Use a calculator to find: \( x \approx \frac{\ln(86.8)}{\ln(7.26)} = \frac{4.463}{1.983} \approx 2.25 \).

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Key Concepts

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

Logarithms
Logarithms, often just referred to as 'logs', are the inverse operations to exponentiation. This means that they undo the exponential process. For instance, if you have an equation like \( a^b = c \), then the logarithm base \( a \) of \( c \) is \( b \), which we write as \( \log_a{c} = b \).

In practical terms, logs answer the question: 'To what power must we raise this base to get this number?' When dealing with exponential equations such as \( (7.26)^{x} = 86.8 \), taking the logarithm of both sides helps us solve for the unknown exponent \( x \). Logs have various bases, the common being base 10, which is the 'common logarithm', and base \( e \), which is the 'natural logarithm'.
Natural Logarithm
The natural logarithm, denoted as \( \ln \), is a specific type of logarithm with a base of \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. The natural log is particularly useful in dealing with growth and decay problems, such as compound interest, population growth, or radioactivity.

It is also extensively used in calculus due to its nice properties when differentiating and integrating. When solving the equation \( (7.26)^{x} = 86.8 \), the natural logarithm is used, and we write the equivalent as \( x = \ln(86.8) / \ln(7.26) \), significantly due to these nice mathematical properties, making calculations less complex.
Power Rule of Logarithms
The power rule of logarithms is a pivotal property that states: when you take the logarithm of a number raised to a power, the exponent can come down to the front as a multiplier. Technically, for any base \( a \) (where \( a > 0 \) and \( a eq 1 \)) and any positive number \( b \), the rule is expressed as \( \log_a(b^c) = c \cdot \log_a(b) \).

Applying this to our problem, we use the natural logarithm to bring down the exponent \( x \) in front of the log, translating \( \ln((7.26)^x) \) to \( x \cdot \ln(7.26) \), simplifying the process of solving equations with exponents. This power rule greatly demystifies handling complex exponential equations by breaking them down into more manageable log-based arithmetic.

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Most popular questions from this chapter

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\log \left(x^{2}-4\right)-1=\log (x+2)$$

If an amount \(a\) is invested at a compound interest rate \(n,\) it will be possible to withdraw a sum \(R\) at the end of every year for \(t\) years until the deposit is exhausted. The number of years is given by $$t=\frac{\log \left(\frac{a n}{R-a n}+1\right)}{\log (1+n)} \quad \text { (years) }$$ If \(\$ 200,000\) is invested at \(12 \%\) interest, for how many years can an annual withdrawal of \(\$ 30,000\) be made before the money is used up?

Simplify each expression. $$\log _{10} 10$$

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$-0.638$$

We can get an approximate value for \(e\) from the following infinite series:$$\begin{aligned} &\begin{array}{l}\text { Series } \\\\\text { Approximation } \quad e=2+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots \quad 158 \\\\\text { for }e\end{array}\\\&\end{aligned}$$.where \(4 !\) (read " 4 factorial") is \(4(3)(2)(1)=24\). Write a program to compute \(e\) using the first five terms of the series.
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