Question

Solve for \(x\).\(7^{x}=\frac{1}{49}\)

Short answer

The solution to the equation is \(x = -2\).

Step-by-step solution

Rewrite the fraction

The first step is to rewrite the fraction \(\frac{1}{49}\) as \(7^{-2}\) because \(\frac{1}{49}\) equals the same value as \(7^{-2}\).

Set the exponents equal to each other

Next, set the exponents in the equation equal to each other. In this case, \(x\) and \(-2\) are the exponents. So a new equation \(x = -2\) can be formed.

State the solution for \(x\)

The solution for \(x\) has already been obtained. So, check if the solution works in the original equation by substituting the value of \(x\) back into the equation. This will confirm whether the solution fits in the equation or not.

Key concepts

Exponents

Exponents are a mathematical shorthand to express repeated multiplication of the same number, known as the base, raised to a power, which represents the number of times to multiply the base by itself. In the equation \(7^x = \frac{1}{49}\), '7' is the base and 'x' is the exponent. Understanding exponents is crucial because they are prevalent in various fields including science, finance, and engineering.

A negative exponent, such as \(-2\), indicates that the base should be taken as the reciprocal and multiplied by itself a positive number of times. For instance, \(7^{-2}\) corresponds to \(\frac{1}{7^2}\), which is \(\frac{1}{49}\). When solving exponential equations, we want to represent both sides of the equation with the same base and exponent if possible, just like the step-by-step solution we see with \(7^x = 7^{-2}\), facilitating a simplified solution.

Inverse Operations

Inverse operations are mathematical operations that reverse the effect of another operation. They are foundational for solving equations. For example, addition and subtraction are inverse operations, just as multiplication and division are. Exponents have their own inverse operation called roots.

When solving the equation \(7^x = \frac{1}{49}\), we use the principle of inverse operations to rewrite \(\frac{1}{49}\) as an expression with the same base of the opposite side: \(7^x = 7^{-2}\). By identifying that raising a number to an exponent and taking a root are inverse operations, we can equate the exponents of the same base directly. This leads to the simple algebraic equation \(x = -2\), bypassing the need for more complicated operations.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. While exponential functions deal with the question 'The base raised to what power gives me this number?', logarithms ask 'To what power must I raise the base to get this number?'. They are denoted as \(\text{log}_b (x)\), which reads as 'log base \(b\) of \(x\)'.

When solving exponential equations where the bases can't be easily made the same, or when dealing with more complex equations, logarithms become an essential tool. For instance, if the given problem was more complex, such as \(7^x = 20\), the use of logarithms would be necessary. You would take the logarithm of both sides, resulting in \(x = \text{log}_7 (20)\), as a way to isolate \(x\) and solve the equation. Recognizing when and how to apply logarithmic functions is a key skill in managing exponential equations.