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Chapter 4: Problem 85

Let \(f(x)=2^{x}, g(x)=(1 / 3)^{x}, h(x)=10^{x},\) and \(m(x)=e^{x} .\) Find the value of \(x\) in each equation. $$g(x)=1$$

Short Answer

Expert verified
x = 0

Step by step solution

01

Understand the given function

The given function is \(g(x) = (\frac{1}{3})^{x}\). To find the value of \(x\) when \(g(x) = 1\), we need to solve \( (\frac{1}{3})^{x} = 1 \).
02

Set up the equation

To solve the equation \((\frac{1}{3})^{x} = 1\), recognize that any number to the power of 0 equals 1. Therefore, we set \(x = 0\).
03

Check the solution

Substitute \(x = 0\) back into the function to verify. \((\frac{1}{3})^{0} = 1\), which confirms the solution is correct.

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Key Concepts

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

Exponential Equations
Exponential equations involve variables in the exponent. They take the form of \(a^{x} = b\). For example, in the problem \(g(x) = \left( \frac{1}{3} \right)^{x} = 1\), we need to find the value of x. Exponential functions grow rapidly, and understanding how to manipulate them is crucial in solving these equations. It's essential to recognize that exponential equations can often be simplified using properties of exponents. Use these properties to rewrite the equation in a simpler form, which allows finding the solution easier. Such skills are particularly helpful in math problems involving growth and decay rates, such as in finance and biology.
Solving for x
Solving for x in exponential equations involves isolating the variable on one side of the equation. For instance, take the equation \((\frac{1}{3})^{x} = 1\). Remember that any non-zero number raised to the power of 0 equals 1. Therefore, we can directly set x = 0. Checking the solution is vital to confirm correctness. Substitute x = 0 back into the equation and verify: \((\frac{1}{3})^{0} = 1\) truly holds. This step-by-step approach ensures no mistakes are made and gives confidence in the result. Sometimes, solving might involve logarithms if the exponents cannot be a simple comparison.
Properties of Exponents
Understanding the properties of exponents is crucial for solving exponential equations. Here are some key properties that will help:
  • \(a^{0}=1\) for any non-zero number 'a'.
  • \(a^{m} * a^{n} = a^{m+n}\).
  • \((a^{m})^{n} = a^{m*n}\).

These properties allow us to manipulate and simplify exponential expressions. In our example, recognizing \(\left( \frac{1}{3} \right)^{0} = 1\) simplifies solving for x. Practicing these properties will solidify your understanding and make solving exponential equations quicker and easier.

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Most popular questions from this chapter

Group Toss Have all students in your class stand and toss a coin. Those that get heads sit down. Those that are left standing toss again and all who obtain heads must sit down. Repeat until no one is left standing. For each toss record the number of coins that are tossed- for example, \((1,30),(2,14),\) and so on. Do not include a pair with zero coins tossed. Enter the data into a graphing calculator and use exponential regression to find an equation of the form \(y=a \cdot b^{x}\) that fits the data. What percent of those standing did you expect would sit down after each toss? Is the value of \(b\) close to this number?

Power Rule Applying the power rule to \(y=\log \left(x^{2}\right)\) yields \(y=2 \cdot \log (x),\) but are these functions the same? What is the domain of each function? Graph the functions. Find another example of a function whose domain changes after application of a rule for logarithms.

Find the amount when \(\$ 10,000\) is invested for 5 years and 3 months at \(4.3 \%\) compounded continuously.

Economic Impact An economist estimates that \(75 \%\) of the money spent in Chattanooga is respent in four days on the average in Chattanooga. So if \(P\) dollars are spent, then the economic impact \(I\) in dollars after \(n\) respendings is given by $$I=P(0.75)^{n}$$ When \(I <0.02 P,\) then \(P\) no longer has an impact on the economy. If the Telephone Psychics Convention brings \(\$ 1.3\) million to the city, then in how many days will that money no longer have an impact on the city?

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.
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