Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the logarithmic equation. $$\log _{x} 81=2$$

Step-by-Step Solution

Answer

The solution to the equation \( \log_{x} 81 = 2 \) is \( x = 9 \) and \( x = -9 \)

Step 1: Understand the logarithm

In the equation \( \log_{x} 81 = 2 \), we wish to find what number (x) must be squared (because 2 is our exponent) to get 81.

Step 2: Convert to exponential form

The logarithmic equation can be converted to an exponential form to make solving easier. The conversion follows this pattern: if \( \log_{b} a = c \), then \( b^c = a \). So, our equation \( \log_{x} 81 = 2 \) becomes \( x^2 = 81 \).

Step 3: Solve for x

Finally, solve the equation \( x^2 = 81 \) for x. This gives two possible solutions: \( x = 9 \) and \( x = -9 \)

Key Concepts

Exponential Form

Converting logarithmic equations into exponential form helps us see the problem differently, which can make it easier to solve. When you see a logarithmic equation like \( \log_{x} 81 = 2 \), it essentially means "what base \( x \) raised to the power of 2 would result in 81?" This might seem a bit confusing with words, but converting it to an exponential form clear things up.

By using the conversion rule from logarithms to exponential form, if \( \log_{b} a = c \), then \( b^c = a \), you can translate the logarithmic equation into \( x^2 = 81 \). This conversion is pivotal because now we have an equation that is usually more familiar and easier to manipulate with basic algebraic techniques.

In exponential form, the equation is all about finding the base number (\( x \)) when raised to the power of 2 gives us the number 81.

Solving Equations

After converting the logarithmic equation into the exponential form, solving the equation \( x^2 = 81 \) becomes a straightforward task. Solving such equations involves finding the values of \( x \) that satisfy this equality.

Here's how to solve \( x^2 = 81 \):

• You need to think about what numbers, when squared, will give the result 81.
• The simple method is to take the square root of both sides of the equation.

By doing so, you find two possible solutions because both a positive and a negative number can be squared to give a positive product. So, we get:

• \( x = 9 \)
• \( x = -9 \)

These two values, 9 and -9, are the solutions to the equation. A positive or a negative number squared will still yield a positive result, fitting the equation \( x^2 = 81 \).

Inverse Operations

Inverse operations are the key to unraveling many mathematical equations. The concept of inverse operations is essentially about "undoing" what has been done.

In the context of logarithmic equations, finding inverse operations helps to convert between forms or to solve equations. Specifically for our problem, finding the square root is the inverse operation used to solve the equation \( x^2 = 81 \).

When you are given \( x^2 = 81 \), you perform the inverse operation of squaring, which is taking the square root. This rules the operations back and allows you to find the numbers that \( x \) could be:

• \( x = \sqrt{81} = 9 \)
• \( x = -\sqrt{81} = -9 \)

By using inverse operations, you effectively reverse the mathematical steps, enabling you to find the possible values of \( x \) that solve the equation.