Problem 94 Solve each equation. $$ \log... [FREE SOLUTION]

Chapter 6: Problem 94

Solve each equation. $$ \log _{x}\left(\frac{1}{8}\right)=3 $$

Short Answer

Expert verified

x = $ \frac{1}{2} $

Step by step solution

Understanding the logarithmic equation

Rewrite the given equation \ \ \[ \log _{x}\frac{1}{8}=3 \ \ \] as an exponential equation. Remember, \ \[ \log _{b} a = c \] can be rewritten as \[ a = b^c \].

Rewriting in exponential form

Using the property of logarithms, rewrite \ \[ \log _{x}\frac{1}{8}=3 \ \ \] to its exponential form: \ \[ \frac{1}{8}=x^3 \].

Solving for x

To find the value of \ x \, solve the equation \ \[ \frac{1}{8} = x^3 \]. Take the cube root of both sides to get: \ \[ x = \frac{1}{2} \].

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Key Concepts

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

logarithms

Logarithms are a way to solve problems involving exponents.
They help us find the power to which a base number is raised to get a particular value.
The basic form of a logarithm is $ \log_b a = c $, where $ b $ is the base, $ a $ is the value, and $ c $ is the exponent.
For example, if we have $ \log_2 8 = 3 $, it means 2 must be raised to the power of 3 to get 8.
Practicing with logarithms can make it easier to understand exponential growth or decay.

exponential form

One crucial property of logarithms is that they can be converted into exponential form.
This is useful for solving logarithmic equations. The transformation rule is:
\[\log_b a = c \ \rightarrow a = b^c.\]
By applying this rule to our original equation $ \log_x \left( \frac{1}{8} \right) = 3 $, we rewrite it as:
\[\frac{1}{8} = x^3.\]
This makes the equation easier to solve because it's now in a form many find more familiar.
Understanding this property helps in both algebra and calculus.

cube root

A cube root is a number that, when multiplied by itself three times, gives the original number.
For instance, the cube root of 8 is 2 because $2 \times 2 \times 2 = 8.$
In our problem, we have the equation $ x^3 = \frac{1}{8} $.
To find $ x $, we need the cube root of $ \frac{1}{8} \.$
Since $ 2 \times 2 \times 2 = 8$, it follows that $ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.
Thus, $ x = \frac{1}{2}$.
Understanding cube roots is key for solving many types of equations in math.

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91影视

Solve each equation. $$ \log _{x}\left(\frac{1}{8}\right)=3 $$

Short Answer

Step by step solution

Understanding the logarithmic equation

Rewriting in exponential form

Solving for x

Key Concepts

logarithms

exponential form

cube root

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