Question

Solve for \(x\).\(2^{x}=64\)

Short answer

Applying the properties of exponents to the given equation, it is found that \(x=6\).

Step-by-step solution

Express the number on the right side of the equation as a power of the base number on the left side

Here, the right side of the equation is 64. The base of the term on the left side is 2, so express 64 as a power of 2. \(64 = 2^{6}\) because 2 multiplied by itself 6 times equals 64.

Use the property that if a^m = a^n, then m = n

Now that both sides of the equation have the same base, you can equate the exponents. \(x = 6\).

Confirm solution

Check the solution by substituting \(x\) with 6 in the original equation \(2^{x}=64\). The left side becomes \(2^{6}\), which equals 64, confirming that \(x=6\) is the solution.

Key concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. The exponent signifies how many times the base is multiplied by itself. For instance, in the expression \(2^{3}\), 2 is the base and 3 is the exponent, indicating that 2 should be multiplied by itself 3 times, resulting in \(2 \times 2 \times 2 = 8\).

Understanding exponentiation is fundamental when it comes to solving exponential equations, since the operation is at the heart of these equations. It is especially important to recognize the relationship between the base and the exponent to be able to manipulate and simplify the expressions effectively.

As an improvement advice, remembering familiar powers of numbers helps when trying to express a number like 64 in terms of a smaller base like 2. Practice recalling powers of 2, for instance, can be immensely helpful in quickly recognizing that \(2^{6}=64\) without needing to calculate each multiplication step by step.

Exponential Functions

Exponential functions are a type of mathematical function involving exponentiation where the variable appears in the exponent. An exponential function follows the form \(f(x) = a^{x}\), where \(a\) is a constant base that is raised to the power of \(x\), the independent variable. In the exercise \(2^{x}=64\), the left side \(2^{x}\) is an example of an exponential function.

These functions are known for their unique properties, particularly their rates of growth which increase rapidly. They have widespread applications in fields such as finance, computer science, and natural sciences. For example, phenomena such as radioactive decay and population growth can be modeled using exponential equations.

An essential strategy for solving an exponential function, like in our exercise, is to manipulate the equation so that both sides have the same base, thus enabling us to equate the exponents.

Equating Exponents

Equating exponents is a method that is used to solve exponential equations where two exponential expressions are set equal to each other. When the base of these expressions is the same, the corresponding exponents must be equal for the equality to hold true.

This concept is based on the property that if \(a^{m} = a^{n}\), then \(m = n\), provided that the base \(a\) is the same and \(a \eq 0\). This technique is effectively utilized in our example of \(2^{x}=64\). Once we recognize that 64 can be expressed as \(2^{6}\), the equation simplifies to \(2^{x} = 2^{6}\), and we can deduce that \(x = 6\).

This method neatly bypasses the need for more complex algebraic manipulations, offering a straightforward route to the solution. It's also a reminder of why understanding the relationship between powers and their roots is key in solving exponential equations. Students should practice this technique with different bases to master the concept.