Problem 14 - Precalculus with Limits

Question

Solve for \(x.\) \(\log _{5} x=\frac{1}{2}\)

Step-by-Step Solution

Answer

The solution to the equation \( \log_{5} x = 1/2 \) is \(x = \( \sqrt{5} \) \), which is approximately equal to \(2.23607\).

Step 1: Convert the Logarithmic Equation to an Exponential Equation

First, the logarithmic equation \(\log _{5} x=\frac{1}{2}\) should be converted into an exponential equation. According to the logarithm rules, the equation \(\log _{b} a = n \) is equivalent to \(b^n = a \). Using this rule, the equation \(\log _{5} x=\frac{1}{2}\) can be converted to \(5^{1/2} = x\).

Step 2: Calculate the Value of x

Now from the resulting exponential equation in the first step \(5^{1/2} = x\), we can calculate the square root of 5 to find the value of \(x\). So, \(x=\sqrt{5}\) which approximately equals to \(x = 2.23607\) when rounded to five decimal places.

Key Concepts

Exponential Equations

Exponential equations involve expressions where variables appear as exponents. In these equations, the same base appears with different exponents, making it crucial to understand how to manipulate and solve them. To solve a logarithmic equation by converting it to an exponential one, you need to understand the relationship between logarithms and exponents.

• An equation of the form \(\log_b a = n\) can be converted to its exponential form \(b^n = a\).
• This conversion is helpful because it allows one to isolate the variable more easily and solve for it directly.
• In the context of the problem, converting \(\log_5 x = \frac{1}{2}\) to \(5^{1/2} = x\) is key to simplifying the equation.

Once converted to the exponential form, solving the equation becomes a straightforward task of evaluating the right side of the equation.

Logarithmic Rules

Logarithmic equations can be understood better through various rules that govern their operations. Knowing these rules is fundamental for transforming and solving equations.

• One of the primary rules is the conversion rule, which states that a logarithm \(\log_b a = n\) can rearrange to become an exponential equation \(b^n = a\).
• Other important rules include the product rule, the quotient rule, and the power rule for logarithms.
• The base rule: changing the base of a logarithm can also help simplify an equation or solve it more easily.

These rules serve as tools to manipulate logarithmic expressions and solve equations effectively, allowing for a more systematic approach to tackling problems involving logarithms.

Square Root Calculations

Square root calculations often come into play when solving exponential equations. Understanding square roots is vital, particularly in cases where you convert the logarithmic equation into an exponential one and need to find the value of a variable.

• The square root of a number \(a\) is a value which, when multiplied by itself, gives \(a\). So, \(\sqrt{5}\) is the number that, when squared, results in 5.
• When evaluating square roots for non-perfect squares, approximations or decimal values are often used. For example, \(\sqrt{5} \approx 2.23607\).
• These calculations are straightforward but require familiarity with basic arithmetic and sometimes a calculator for precision.

Understanding these simple calculations is crucial, as they are frequent in mathematical problem-solving, including solving exponential and logarithmic equations.