Problem 74 - Precalculus : Building Concepts and Connections

Question

Solve each exponential equation. $$ 2^{x-1}=10 $$

Step-by-Step Solution

Answer

\[\log_{2}10 + 1 = x\]. This is the value of \(x\)

Step 1: Convert The Exponential Equation Into A Logarithmic Equation

An exponential equation can be converted into a logarithmic equation. The given equation \(2^{x-1} = 10\) can be rewritten in logarithmic form. The base of the exponent becomes the base of the logarithm, the exponent becomes the argument of the logarithm, and the value that the exponential equation is set equal to becomes the value that the logarithmic equation is set equal to. So, the equation can be rewritten as log base 2 of 10 equals \(x-1\), or \(\log_{2}10 = x - 1\).

Step 2: Solve for x

Next, isolate the variable \(x\) by adding 1 to both sides of the equation. This gives \(\log_{2}10 + 1 = x\).

Step 3: Simplify

Since the base 2 logarithm of 10 isn't a simple number, we leave it in that form and add one to get the final result: \(\log_{2}10 + 1 = x\).

Key Concepts

Logarithmic Equations

Logarithmic equations often appear when you are working to solve exponential equations. These are equations that involve a logarithm, which is the inverse operation of exponentiation. Solving logarithmic equations typically involves using properties of logarithms to simplify the expression or convert it to an exponential form in order to solve for the unknown variable.

• When confronted with an exponential equation, converting it into a logarithmic equation can make the solution more manageable.
• This transformation leverages the inverse operations of logarithms and exponents, simplifying complex exponential relationships.

A crucial step in dealing with these equations is recognizing the properties of logarithms such as product, quotient, and power rules, which can transform the equation into solvable terms. For instance, if you have an equation like \(2^{x-1} = 10\), converting it into a form such as \(\log_{2}10 = x - 1\) paves the way to solve it systematically through further operations.

Logarithmic Form

Understanding the logarithmic form of an equation is essential when solving exponential equations. The logarithmic form allows you to restate an exponential equation in a way that makes finding the solution more straightforward. The general form of converting an exponential equation \(b^y = x\) into its logarithmic form is \(\log_b x = y\).

• With the considered equation \(2^{x-1} = 10\), rewriting in logarithmic form involves recognizing the base 2 and expressing the equation as \(\log_{2}10 = x - 1\).
• This form provides clarity by isolating the part of the equation that involves the variable \(x\), which is advantageous when rearranging and solving the equation.

Adopting this form is particularly useful when the equation does not resolve to whole numbers or known log values because it allows leaving stuck terms in logarithmic notation. This might not give a decimal result immediately but sets up for numerical calculation.

Isolating Variables

Isolating variables is a fundamental skill for solving equations in any mathematical discipline. The ultimate goal is to rearrange the equation such that the variable we are solving for stands alone on one side of the equation.

• In our specific logarithmic equation \(\log_{2}10 = x - 1\), isolating \(x\) requires us to perform algebraic operations that place \(x\) on one side by itself.
• This is accomplished by adding 1 to both sides of the equation, turning it into \(x = \log_{2}10 + 1\).

Isolating variables is vital for finding solutions, converting forms, and understanding the relationships between components of mathematical equations. Mastering this process enables easier manipulation of equations and facilitates checking the validity of solutions by back-substitution.