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Chapter 10: Problem 56

Solve each equation. $$\log _{x} \frac{1}{10}=-1$$

Short Answer

Expert verified
x = 10

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\log_{x} \frac{1}{10} = -1\). This equation tells us that when x is raised to the power of -1, the result is \(\frac{1}{10}\).
02

Convert the logarithmic form to exponential form

Recall that \(\log_{b}(a) = c\) is equivalent to \(b^{c} = a\). Using this identity, convert the given equation into exponential form. In this case, \(x^{-1} = \frac{1}{10}\).
03

Simplify the exponential equation

The equation \(x^{-1} = \frac{1}{10}\) can be rewritten as \(\frac{1}{x} = \frac{1}{10}\), which implies that \(x = 10\).
04

Verify the solution

To confirm that the solution is correct, substitute \(x = 10\) back into the original logarithmic equation. \(\log_{10} \frac{1}{10}\) indeed equals \(-1\), which verifies that our solution is correct.

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Key Concepts

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

logarithmic form
Let's start by understanding what logarithms are. Logarithmic form gives us a way to express exponents. If you see an equation like \(\text{log}_{b}(a) = c\), it reads as: 'the logarithm of \(a\) with base \(b\) is \(c\).'
Essentially, it鈥檚 a different way of asking 'to what power must we raise \(b\) to get \(a\)?'
In our exercise, we have \(\text{log}_{x}\frac{1}{10} = -1\).
Here, \(x\) is the base, \(\frac{1}{10}\) is the number we鈥檙e taking the log of, and \(-1\) is the result of the log.
This formulation helps us solve the equation.
exponential form
Now, let鈥檚 talk about converting logarithmic form to exponential form.
This conversion is key in solving logarithmic equations.
Remember, \(\text{log}_{b}(a) = c\) can be rewritten as \[ b^{c} = a \].
This means: 'b (the base) raised to the power of c equals a.'
Applying this to our original equation \(\text{log}_{x}\frac{1}{10} = -1\), we convert it into the exponential form: \[ x^{-1} = \frac{1}{10} \].
This switch simplifies our work, making it easier to solve for the unknown variable \(x\).
By understanding both forms, we can easily move between logarithmic and exponential equations.
solving logarithmic equations
Let's put it all together and solve the equation \(\text{log}_{x}\frac{1}{10} = -1\).
Following our steps:
- Convert to exponential form: \[ x^{-1} = \frac{1}{10} \]
- Simplify the exponential equation: \[ \frac{1}{x} = \frac{1}{10} \]
- Both sides are fractions, and we can directly see \(x = 10\).
It's always good practice to double-check our solution by substituting back into the original equation.
If we replace \(x\) with \(10\) in \(\text{log}_{10} \frac{1}{10} = -1\), we see the equation holds true, confirming our solution \(x = 10\).
Understanding these steps makes solving logarithmic equations straightforward.

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Most popular questions from this chapter

Solve each equation. Approximate solutions to three decimal places. $$ 4^{x-2}=5^{3 x+2} $$

Determine whether common logarithms or natural logarithms would be a better choice to use for solving each equation. Do not actually solve. $$ 10^{0.0025 x}=75 $$

The concentration of a drug in a person's system decreases according to the function $$ C(t)=2 e^{-0.125 t} $$ where \(C(t)\) is in appropriate units, and \(t\) is in hours. Approximate answers to the nearest hundredth. (a) How much of the drug will be in the system after \(1 \mathrm{hr} ?\) (b) How long will it take for the concentration to be half of its original amount?

Based on selected figures obtained during the years \(1970-2015,\) the total number of bachelor's degrees earned in the United States can be modeled by the function $$ D(x)=792,377 e^{0.01798 x} $$ where \(x=0\) corresponds to \(1970, x=5\) corresponds to \(1975,\) and so on. Approximate, to the nearest unit, the number of bachelor's degrees earned in 2015. (Data from U.S. National Center for Education Statistics.)

The population of deer (in thousands) in a certain area is approximated by the logarithmic function $$f(x)=\log _{5}(100 x-75)$$where \(x\) is the number of years since 2017 . During what year is the population expected to be 4 thousand deer?
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