Problem 67 - Elementary and Intermediate Algebra

Question

Solve for \(x\). See Example 3 . $$ \log _{100} x=\frac{3}{2} $$

Step-by-Step Solution

Answer

\( x = 1000 \)

Step 1: Understand the Logarithmic Equation

We start with the equation \( \log _{100} x = \frac{3}{2} \). This means that 100 raised to the power of \( \frac{3}{2} \) equals \( x \). In other words, we need to find the value of \( x \) such that this equation holds true.

Step 2: Convert to Exponential Form

Since the logarithmic equation \( \log _{100} x = \frac{3}{2} \) implies \( x = 100^{\frac{3}{2}} \), we need to convert the equation into its exponential form. This gives us the expression \( x = 100^{\frac{3}{2}} \).

Step 3: Simplify the Exponential Expression

To find \( 100^{\frac{3}{2}} \), recognize that \( 100 \) can be rewritten as \( 10^2 \). Thus, \( 100^{\frac{3}{2}} = (10^2)^{\frac{3}{2}} = 10^{2 \times \frac{3}{2}} = 10^3 \).

Step 4: Calculate the Power

Now, calculate \( 10^3 \). Since \( 10^3 = 10 \times 10 \times 10 = 1000 \), we find that \( 100^{\frac{3}{2}} = 1000 \).

Step 5: Conclude the Value of x

After simplifying and calculating the expression, we find that \( x = 1000 \). Therefore, the solution to the equation \( \log _{100} x = \frac{3}{2} \) is \( x = 1000 \).

Key Concepts

Exponential Form

The exponential form of a logarithmic equation provides a way to express the relationship between the base, exponent, and result without using logarithms. It is particularly helpful when solving logarithmic equations like \( \log_{100} x = \frac{3}{2} \). Here, we convert the logarithmic equation into the exponential form:

• The base \( 100 \) becomes the base of the exponent.
• The right side of the equation, \( \frac{3}{2} \), becomes the exponent.
• The result equals the number the log originally operated on, which is \( x \).

So, the equation \( \log_{100} x = \frac{3}{2} \) in exponential form is written as \( x = 100^{\frac{3}{2}} \). This means that \( 100 \) is raised to the power \( \frac{3}{2} \) resulting in \( x \). Having this equation in exponential form makes it more intuitive to solve, as you can now focus on simplifying and calculating the power.

Simplifying Exponents

To simplify the expression \( 100^{\frac{3}{2}} \), it’s beneficial to break down the base into a simpler form. Since \( 100 \) can be expressed as \( 10^2 \), this allows us to rewrite the expression:

• \( 100^{\frac{3}{2}} \) becomes \((10^2)^{\frac{3}{2}} \).
• When simplifying exponents with a power to a power, multiply the exponents: \( 10^{2 \times \frac{3}{2}} \).
• Calculate the multiplication: \(2 \times \frac{3}{2} = 3 \).

So, \( 100^{\frac{3}{2}} \) simplifies down to \( 10^3 \). The property used here, \((a^m)^n = a^{m\times n}\), is fundamental in simplifying exponents. By reducing it to \( 10^3 \), the calculation process is made easier.

Power Calculations

Once you have simplified the expression to a simpler power, as in \( 10^3 \), you're ready to perform the power calculations. Calculating powers involves multiplying the base by itself as many times as specified by the exponent:

• \( 10^3 \) means multiplying the number 10 three times: \( 10 \times 10 \times 10 \).
• Start by calculating \( 10 \times 10 = 100 \).
• Then, multiply the result by 10: \( 100 \times 10 = 1000 \).

Thus, \( 10^3 = 1000 \). This shows that simplifying the base and exponent relationship makes the power calculation straightforward. Concluding, since \( x = 100^{\frac{3}{2}} = 1000 \), the solution of the original logarithmic equation is \( x = 1000 \). This step of calculating powers is a crucial final step in solving exponential equations derived from logarithms.