Problem 45 - Algebra 2

Question

Solve each equation. Check your solutions. \(\log _{25} n=\frac{3}{2}\)

Step-by-Step Solution

Answer

The solution is \( n = 125 \).

Step 1: Understand the Given Equation

The equation \( \log_{25} n = \frac{3}{2} \) means we need to find the value of \( n \) such that 25 raised to the power of \( \frac{3}{2} \) equals \( n \).

Step 2: Convert the Logarithmic Equation to Exponential Form

To convert the equation \( \log_{25} n = \frac{3}{2} \) to an exponential form, we rewrite it as: \( 25^{\frac{3}{2}} = n \). This tells us that 25 raised to the power of \( \frac{3}{2} \) is equal to \( n \).

Step 3: Simplify the Exponential Expression

To compute \( 25^{\frac{3}{2}} \), first take the square root of 25, which is 5, and then raise it to the power of 3. Thus, \( 25^{\frac{3}{2}} = (\sqrt{25})^3 = 5^3 = 125 \). Therefore, \( n = 125 \).

Step 4: Check the Solution

To verify our solution, we substitute \( n = 125 \) back into the original logarithmic equation. Calculate \( \log_{25} 125 \). If the result is \( \frac{3}{2} \), the solution is correct. Since \( 25^{\frac{3}{2}} = 125 \), the value checks out, confirming our solution is correct.

Key Concepts

Exponential Form

The logarithmic equation \( \log_{25} n = \frac{3}{2} \) can be quite confusing at first glance. But, when broken down, this simply means that \( 25 \) raised to a certain power gives us \( n \). The term "exponential form" refers to the expression of a number as a base raised to a power.
In our exercise, we want to convert the logarithmic equation into an easier-to-understand exponential form. We do this by rewriting the equation as \( 25^{\frac{3}{2}} = n \).

• Here, \( 25 \) is our base.
• \( \frac{3}{2} \) is the exponent.
• \( n \) is the result or product of this exponentiation.

Understanding this conversion is essential as it transforms a logarithmic equation into a form where calculations are more straightforward.

Simplifying Exponents

Once we have rewritten our equation in exponential form \( 25^{\frac{3}{2}} = n \), the next step is simplifying exponents. This process involves breaking the exponent into manageable parts to easily perform the calculation.
The exponent \( \frac{3}{2} \) can be understood as taking the square root of 25 first, then raising the result to the power of 3. Let's break it down step-by-step:

• Calculate the square root of 25: \( \sqrt{25} = 5 \)
• Next, raise the result (5) to the power of 3: \( 5^3 = 125 \)

Thus, \( 25^{\frac{3}{2}} = 125 \).
This shows how careful breakdown of exponents yields a comprehensible outcome, which in our case is \( n = 125 \). Understanding this simplification helps to relate each part of the exponent to its corresponding algebraic operation.

Verification of Solutions

After determining a solution, it's crucial to verify that it is correct by substituting it back into the original equation. Verifying our solution assures us that it's both reasonable and accurate.
In our given problem, we found that \( n = 125 \). Verification involves checking whether this value satisfies the original logarithmic equation \( \log_{25} 125 = \frac{3}{2} \). We do this by:

• Calculating \( 25^{\frac{3}{2}} \) once more, which we've established equals 125.
• Comparing if the expression \( \log_{25} 125 \) indeed equals \( \frac{3}{2} \).

Since both computations align, the solution \( n = 125 \) is confirmed to be accurate. This process of verification emphasizes the importance of validating our results to ensure they meet the problem’s initial conditions.