Problem 57 - Intermediate Algebra

Question

Solve each equation. $$ \log 5-\log x=1 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{2}\).

Step 1: Apply Logarithmic Identity

Use the logarithmic identity \[\log a - \log b = \log \left(\frac{a}{b}\right)\]to combine the logarithms on the left-hand side of the equation:\[\log 5 - \log x = \log \left(\frac{5}{x}\right)\]So the equation becomes:\[\log \left(\frac{5}{x}\right) = 1\]

Step 2: Convert Logarithmic Equation to Exponential Form

Recall that if \(\log_b a = c\), then the equation can be rewritten in exponential form as \(a = b^c\). Here, the base is 10 (common logarithm), so\[\frac{5}{x} = 10^1\]which simplifies to\[\frac{5}{x} = 10\]

Step 3: Solve for x

Cross-multiply to solve for \(x\):\[5 = 10x\]Divide both sides by 10 to isolate \(x\):\[x = \frac{5}{10}\]Simplify the fraction:\[x = \frac{1}{2}\]

Key Concepts

Logarithmic Identities

Logarithmic identities are key tools in simplifying and solving logarithmic equations. In this exercise, we use the identity \(\log a - \log b = \log \left(\frac{a}{b}\right)\). This identity helps you combine two logarithms into a single logarithm.
It’s derived from properties of exponents and is useful when dealing with subtraction of two logarithms.

• This identity effectively condenses multiple logarithmic operations into one.
• It can simplify the equation, making it easier to solve.
• This concept can be extended to other bases beyond the common logarithm (base 10).

Always remember: when subtracting logs with the same base, you can condense them using division inside the single log term. This not only streamlines calculations but also helps in maintaining clarity throughout problem-solving.

Exponential Form

The exponential form of an equation is a way to express the relationship between logarithms and exponents.
In our equation, we start with \(\log \left(\frac{5}{x}\right) = 1\). Knowing the base of the logarithm is 10 (since it's a common logarithm), we can convert this to its exponential form as \(\frac{5}{x} = 10^1\).

• This conversion from logarithmic to exponential form makes solving for variables straightforward.
• It’s like translating the equation into a form that's simpler to work with.
• Remember the general rule: if \(\log_b a = c\), then \(a = b^c\).

Understanding how to switch between these forms enables you to solve various logarithmic and exponential equations efficiently.

Common Logarithms

Common logarithms use the base 10 and are often implied without writing the base explicitly. For instance, \(\log 5\) and \(\log x\) in this exercise are actually \(\log_{10} 5\) and \(\log_{10} x\).
Recognizing the base is key, as it dictates how you proceed with solving the equations.

• Common logarithms are especially useful in scientific and engineering contexts due to their simplicity and relevance to decimal systems.
• They allow for the easy use of logarithmic identities, like the one used initially in this problem.
• Knowing that \(\log_{10} 10 = 1\) makes calculations involving \(10\) very convenient.

Familiarity with common logarithms will help you handle a variety of problems more smoothly, especially if you frequently encounter decimal-based calculations.