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Chapter 4: Problem 77

Write each equation as an equivalent logarithmic equation. $$a^{x-1}=n$$

Short Answer

Expert verified
The equivalent logarithmic equation is \[ \text{log}_a(n) = x-1 \].

Step by step solution

01

Identify the Exponential Form

The given equation is in exponential form: \[ a^{x-1} = n \] This needs to be changed into its logarithmic form.
02

Recall the Definition of Logarithms

The definition of a logarithm states that if \[ a^b = c \] then its logarithmic form is \[ \text{log}_a(c) = b \].
03

Apply the Definition

Using the definition from the previous step, identify the base ( \(a\) ), the exponent ( \(x-1\) ), and the result ( \(n\) ).
04

Rewrite in Logarithmic Form

Replace the values into the logarithmic form equation: \[a^{x-1} = n\] becomes \[ \text{log}_a(n) = x-1 \].

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Key Concepts

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

exponential form
When tackling equations, it often comes down to recognizing their form. The exponential form is expressed as:

\[ a^b = c \]
Here, a is the base, b is the exponent, and c is the result. Understanding this form helps you identify the key components of the equation.

For example, in the given equation \[ a^{x-1} = n \]
- a is the base.- x-1 is the exponent.- n is the result.

Recognizing these parts allows us to transition to a logarithmic form easily.
definition of logarithms
The next important idea is understanding how logarithms work.

The definition of logarithms can be stated as follows:

If you have an exponential equation in the form \[ a^b = c \], then its equivalent logarithmic form is:\[ \text{log}_a(c) = b \]

This definition essentially means that the logarithm asks the question: 'To what power must the base a be raised, to produce the number c?'

This understanding helps us translate between exponential and logarithmic forms. It鈥檚 a powerful tool because sometimes working in the logarithmic form can simplify solving complex equations.
logarithmic form
Given what we know about exponential forms and the definition of logarithms, we can now rewrite equations in their logarithmic form.

Starting with the exponential equation:\[ a^{x-1} = n \]

We identify the base (a), the exponent (x-1), and the result (n).

Using the definition of logarithms, we convert it into its equivalent logarithmic form:
\[ \text{log}_a(n) = x-1 \]
This forms a new way to express the relationship outlined in the original exponential equation. In the logarithmic form, it now directly shows how the result (n) relates back to the base and the exponent.

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Most popular questions from this chapter

Which exponential and logarithmic functions are increasing? Decreasing? Is the inverse of an increasing function increasing or decreasing? Is the inverse of a decreasing function increasing or decreasing? Explain.

To evaluate an exponential or logarithmic function we simply press a button on a calculator. But what does the calculator do to find the answer? The next exercises show formulas from calculus that are used to evaluate \(e^{x}\) and \(\ln (1+x)\). Infinite Series for \(e^{x}\) The following formula from calculus is used to compute values of \(e^{x}\) : $$e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\cdots+\frac{x^{n}}{n !}+\cdots$$ where \(n !=1 \cdot 2 \cdot 3 \cdot \cdots \cdot n\) for any positive integer \(n .\) The notation \(n !\) is read " \(n\) factorial." For example, \(3 !=1 \cdot 2 \cdot 3=6\) In calculating \(e^{x},\) the more terms that we use from the formula, the closer we get to the true value of \(e^{x}\). Use the first five terms of the formula to estimate the value of \(e^{0.1}\) and compare your result to the value of \(e^{0.1}\) obtained using the \(e^{x}-\) key on your calculator.

Visual Magnitude of a Star If all stars were at the same distance, it would be a simple matter to compare their brightness. However, the brightness that we see, the apparent visual magnitude \(m,\) depends on a star's intrinsic brightness, or absolute visual magnitude \(M_{V},\) and the distance \(d\) from the observer in parsecs ( 1 parsec \(=3.262\) light years), according to the formula \(m=M_{V}-5+5 \cdot \log (d) .\) The values of \(M_{V}\) range from \(-8\) for the intrinsically brightest stars to \(+15\) for the intrinsically faintest stars. The nearest star to the sun, Alpha Centauri, has an apparent visual magnitude of 0 and an absolute visual magnitude of 4.39 . Find the distance \(d\) in parsecs to Alpha Centauri.

Cost of a Parking Ticket The cost of a parking ticket on campus is \(\$ 15\) for the first offense. Given that the cost doubles for each additional offense, write a formula for the cost C as a function of the number of tickets \(n\) First write some ordered pairs starting with \((1,15)\)

Solve each problem. Solve the equation \(A=P e^{n t}\) for \(r,\) then find the rate at which a deposit of \(\$ 1000\) would double in 3 years compounded continuously.
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