Problem 101 - College Algebra

Question

Write each equation in its equivalent exponential form. Then solve for \(x .\) $$ \log _{3}(x-1)=2 $$

Step-by-Step Solution

Answer

The equivalent exponential form of the equation is \(3^{2} = x-1\), and by subsequently solving for x yields \(x=10\).

Step 1: Convert Logarithm to Exponential Form

Recall that the equation \(\log _{b}(a)=c\) is equivalent to an exponential form \(b^c=a\). Thus, the equation \(\log _{3}(x-1)=2\) can be written as \(3^{2} = x-1\).

Step 2: Simplify

Calculate \(3^{2}\) to simplify the equation. This gives \(9=x-1\).

Step 3: Solve for x

Finally, add 1 to each side in the equation \(9=x-1\) to obtain x on its own. This results in x = 10.

Key Concepts

Logarithms

Logarithms are mathematical expressions that help us solve exponential equations by transforming them into linear forms. The basic idea is to use the logarithm function to reverse the process of exponentiation.
The core principle of logarithms is that they relate to exponents. If you have an equation of the form \( \log_{b}(a) = c \)this means the base \( b \), when raised to the power of \( c \), equals \( a \).
This can be expressed as \( b^c = a \).

• The base \( b \) is the number being multiplied.
• The exponent \( c \) tells you how many times the base is multiplied by itself.
• The result \( a \) is what you get after the multiplication is complete.

Understanding logarithms is crucial because they simplify working with exponential functions, making it easier to manipulate and solve equations that involve powers.

Solving Equations

Solving equations means finding values for the variables that make the equation true. The solution process often involves manipulating the equation to isolate the unknown variable.
For instance, in the exercise given, we have to solve for \( x \) in the equation \( 9 = x - 1 \).

• To isolate \( x \), you add 1 to each side of the equation.
• This becomes: \( 9 + 1 = x \).
• Simplifying the right-hand side gives \( x = 10 \).

These steps demonstrate the general process of solving equations by performing operations equally on both sides. This method ensures that the original equality of the equation is maintained, leading to the correct solution.

Exponential Form

Converting a logarithmic equation to its exponential form is a critical skill when working with logarithms. This conversion allows us to express the equation differently, often making it simpler to solve.
For the exercise, we start with the logarithmic equation \( \log_{3}(x-1) = 2 \).

• We recognize that this implies an exponential form: \( 3^2 = x - 1 \).
• Thus, the exponent 2 tells us that the base 3 should be multiplied by itself once.
• After evaluating \( 3^2 \), we find that \( 9 = x - 1 \).

Converting to exponential form is immensely helpful because it turns the problem into a form that is more straightforward to understand and solve, using basic algebraic techniques.