Problem 38 - Algebra 2

Question

Solve each equation. Check your answers. $$ \log (x-2)=1 $$

Step-by-Step Solution

Answer

The solution is \( x = 12 \), checked by substituting back into the original equation.

Step 1: Convert From Logarithmic to Exponential Form

The given equation is \( \log (x-2)=1 \). This is in logarithmic form. We can convert this to exponential form. Since it is a common logarithm (base 10 implicitly) and taking into account the property that the base raised to the output equals the input, the equation becomes: \( 10^1 = x - 2 \).

Step 2: Solve for x

Now, solve the equation above for x. This means isolate x by adding 2 to both sides of the equation, which will give: \( x = 10^1 + 2 \). Calculate the sum on the right side to find the value of x.

Step 3: Check the Solution

We need to check the solution to ensure it's valid. Substitute the value of x back into the original equation and verify whether the equation is satisfied. For the check to be valid, the value inside the logarithm should be positive.

Key Concepts

Exponential Form

Converting from logarithmic form to exponential form can be a handy way to simplify and solve equations. Logarithmic equations express the power to which a base number must be raised to produce a given number. In the equation \( \log (x-2) = 1 \), we're dealing with a common logarithm which uses base 10 implicitly. To convert a logarithmic equation to exponential form, remember the rule: the base raised to the logarithm's output equals the input inside the logarithm.

• Base (10) raised to Output (1) equals Input \((x-2)\).

Once converted, the equation \( 10^1 = x - 2 \) is easier to manipulate. Transferring to exponential form helps in visualizing the equation as a basic arithmetic operation and is a crucial step before isolating the variable to find its value.

Common Logarithm

The common logarithm is simply a logarithm with base 10. It is frequently used due to the widespread basis of the decimal system in everyday mathematics and science.
When writing a common logarithm, the base 10 is typically omitted. For example, \( \log(x-2) \) implies \( \log_{10}(x-2) \).

• Base 10 is standard for common logs, allowing for fewer symbols in equations.
• Understanding that the missing base is 10 helps in converting to exponential form and solving equations.

Knowing how to interpret and convert common logarithms will arm you with the skills needed to tackle various real-world and hypothetical logarithmic problems with ease and efficiency.

Checking Solutions

Verifying your solution is a critical step in solving any equation. After solving for \( x \) in a logarithmic equation, you should always check to confirm that your solution actually satisfies the original equation.
For the equation \( \log (x-2) = 1 \), once you solve for \( x \) and find a value, substitute it back into the original equation:

• Plug \( x \) back into \( \log(x-2) \) and make sure the result equals 1.
• Ensure that \( x-2 > 0 \) since the argument of a logarithmic function must be positive.

These checks prevent errors and confirm the validity of your solution, contributing towards a thorough understanding of logarithmic operations and their properties.