Problem 71 - Algebra 2

Question

Write an equivalent exponential equation. $$ \log _{2} 3=x $$

Step-by-Step Solution

Answer

The equivalent exponential equation is \( 2^x = 3 \).

Step 1: Understand the Given Logarithmic Equation

The equation \( \log_{2} 3 = x \) means that 2 is the base of the logarithm and the result of this logarithmic function is \( x \). It represents the power to which the base 2 must be raised to yield 3.

Step 2: Convert the Logarithmic Equation to Exponential Form

The general form for converting a logarithmic equation \( \log_{b} a = c \) to its equivalent exponential form is given by \( b^c = a \). Applying this to our equation \( \log_{2} 3 = x \), we can write: \( 2^x = 3 \).

Step 3: Verify the Conversion

Ensure that the exponential equation \( 2^x = 3 \) correctly represents the original logarithmic equation. This confirms that raising 2 to the power of \( x \) results in the number 3.

Key Concepts

Understanding Logarithmic Equations

Logarithmic equations express the power or exponent needed to raise a base number to obtain another number. In the equation \( \log_{2} 3 = x \), 2 is the base, 3 is the result, and \( x \) is the exponent. This means we are looking for the power to which 2 must be raised to result in 3. Logarithms are essentially the inverse functions of exponentials, allowing us to determine unknown exponents given a base and its power's result.

To effectively handle logarithms, it's important to grasp how they translate between logarithmic and exponential forms. This correlation is key in solving or simplifying equations with unknown variables raised to a power.

Equation Conversion Techniques

Converting logarithmic equations to exponential form is a crucial technique for solving equations. The conversion serves to simplify the problem and make the unknowns more apparent. The general principle to follow is: if \( \log_{b} a = c \), it can be converted to the exponential form \( b^c = a \).

In our example \( \log_{2} 3 = x \), converting this equation involves recognizing that the base (2) raised to the power (\( x \)) equals the result (3), hence \( 2^x = 3 \). This step is essential in solving for the unknown exponent \( x \).

This form of conversion helps in isolating the variable, making it easier to apply further mathematical tools or numerical approximation if necessary.

The Exponential Form

Exponential form equations express numbers as a base raised to a specified power. This format is particularly useful as it makes complex multiplicative calculations clearer. For the equation \( 2^x = 3 \), it tells us directly that 2 raised to some power, sometimes a non-integer, equals 3.

Exponential forms are straightforward to interpret, as they explicitly show the relationship between the base and the resulting number through an exponent. This form is handy for graphing, analyzing growth and decay in natural processes, and computing compound interest, among others.

Exploring Powers and Exponents

Powers and exponents are fundamental concepts in both algebra and calculus. Simply put, an exponent indicates how many times a base number is multiplied by itself. For instance, \( 2^3 \) means 2 is multiplied by itself twice: \( 2 \times 2 \times 2 \).

In exponential equations like \( 2^x = 3 \), the goal is to determine the value of \( x \), which might not be an integer. This often involves either using logarithms to find the exact value or approximating it using a calculator or table.

Powers and exponents are powerful tools for expressing very large or very small numbers compactly, solving equations involving growth and decay, and performing many tasks in computational sciences.