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Chapter 1: Problem 12

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$a=b^{t}$$

Short Answer

Expert verified
\( t = \frac{\ln(a)}{\ln(b)} \)

Step by step solution

01

Understand the Equation

We are given the equation \( a = b^t \). This means \( a \) equals \( b \) raised to the power of \( t \). We need to solve this equation for \( t \).
02

Take the Natural Logarithm on Both Sides

To solve for \( t \), we need to eliminate the exponent. We do this by taking the natural logarithm (ln) of both sides of the equation. Thus, we have: \( \ln(a) = \ln(b^t) \).
03

Apply the Logarithmic Power Rule

The logarithmic power rule states that \( \ln(b^t) = t \cdot \ln(b) \). Apply this rule to the right side: \[ \ln(a) = t \cdot \ln(b) \].
04

Solve for t

To isolate \( t \), divide both sides of the equation by \( \ln(b) \), resulting in: \[ t = \frac{\ln(a)}{\ln(b)} \].

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In the equation \(a = b^t\), the base is \(b\) and the exponent is \(t\). Exponentiation is fundamentally about repetition.
The base, \(b\), is multiplied by itself \(t\) times.
A few key characteristics of exponentiation include:
  • It is not commutative, meaning \(b^t\) is not the same as \(t^b\).
  • It is associative with respect to multiplication, such that \((b^t)^s = b^{(t\cdot s)}\).
  • The exponent can be any real number, which allows for operations involving roots and fractional exponents.
In our problem, \(a\) represents the outcome of raising \(b\) to \(t\). Understanding this relationship is key when solving equations where the variable is in the exponent.
Logarithmic Power Rule
The logarithmic power rule is a useful tool when dealing with equations where the variable appears as an exponent. The rule provides a way to "bring down" the exponent, making the equation easier to solve.
To apply this rule, you start with the logarithm of an exponentiated number, for instance, \(\ln(b^t)\). The rule states:
  • The logarithm of a power, \(b^t\), is the exponent times the logarithm of the base.
This is mathematically expressed as \(\ln(b^t) = t \cdot \ln(b)\).
This transformation is critical as it reduces the problem from an exponential equation to a linear equation, which is far simpler to manipulate. Recognizing and properly applying the logarithmic power rule is a fundamental skill in solving many logarithmic equations.
Solving Logarithmic Equations
Solving logarithmic equations often involves multiple steps and a comprehensive understanding of how logarithms and exponents interact. When you have an equation like \(a = b^t\), you can solve it by following these steps:
  • Take the natural logarithm: Start by taking the \(\ln\) of both sides, transforming it into \(\ln(a) = \ln(b^t)\).
  • Apply the logarithmic power rule: Use this rule to rewrite \(\ln(b^t)\) as \(t \cdot \ln(b)\).
  • Isolate the variable: Solve for \(t\) by dividing both sides by \(\ln(b)\).
This process results in the solution \(t = \frac{\ln(a)}{\ln(b)}\).
Logarithmic equations often appear complex due to their format, but applying the rules consistently will lead you to a solution. Always double-check your steps, as each transformation must be correct to maintain the equation's equality.

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Most popular questions from this chapter

A health club has cost and revenue functions given by \(C=10,000+35 q\) and \(R=p q,\) where \(q\) is the number of annual club members and \(p\) is the price of a one year membership. The demand function for the club is \(q=3000-20 p\) (a) Use the demand function to write cost and revenue as functions of \(p\) (b) Graph cost and revenue as a function of \(p,\) on the same axes. (Note that price does not go above \(\$ 170\) and that the annual costs of running the club reach \(\$ 120,000 .)\) (c) Explain why the graph of the revenue function has the shape it does. (d) For what prices does the club make a profit? (e) Estimate the annual membership fee that maximizes profit. Mark this point on your graph.

The Hershey Company is the largest US producer of chocolate. In \(2011,\) annual net sales were 6.1 billion dollars and were increasing at a continuous rate of \(4.2 \%\) per year. \(^{65}\) (a) Write a formula for annual net sales, \(S,\) as a function of time, \(t,\) in years since 2011 (b) Estimate annual net sales in 2015 (c) Use a graph to estimate the year in which annual net sales are expected to pass 8 billion dollars and check your estimate using logarithms.

You are buying a car that comes with a one-year warranty and are considering whether to purchase an extended warranty for 375 . The extended warranty covers the two years immediately after the one-year warranty expires. You estimate that the yearly expenses that would have been covered by the extended warranty are 150 at the end of the first year of the extension and 250 at the end of the second year of the extension. The interest rate is \(5 \%\) per year, compounded annually. Should you buy the extended warranty? Explain.

Concern biodiesel, a fuel derived from renewable resources such as food crops, algae, and animal oils. The table shows the percent growth over the previous year in US biodiesel consumption. $$\begin{array}{c|c|c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 \\ \hline \text { * growth } & -12.5 & 92.9 & 237 & 186.6 & 37.2 & -11.7 & 7.3 \\ \hline \end{array}$$ (a) According to the US Department of Energy, the US consumed 91 million gallons of biodiesel in 2005 Approximately how much biodiesel (in millions of gallons) did the US consume in \(2006 ?\) In \(2007 ?\) (b) Graph the points showing the annual US consumption of biodiesel, in millions of gallons of biodiesel, for the years 2005 to \(2009 .\) Label the scales on the horizontal and vertical axes.

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$7 \cdot 3^{t}=5 \cdot 2^{t}$$
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