Question

$$\text {Solve the following equations.}$$ $$7^{x}=21$$

Short answer

Question: Solve the equation 7^x = 21, and find the approximate value of x. Answer: The approximate value of x for the equation 7^x = 21 is x ≈ 1.77124.

Step-by-step solution

Write down the given equation

The given equation is: $$7^{x} = 21$$

Take the logarithm of both sides of the equation.

Taking the logarithm (log) of both sides, we get: $$\log(7^{x}) = \log(21)$$

Use the logarithm power rule to simplify the equation.

Using the logarithm power rule (i.e. \(\log(a^{b}) = b\log(a)\)), we get: $$x \log(7) = \log(21)$$

Solve for x.

Now, we solve for x by dividing both sides by \(\log(7)\): $$x = \frac{\log(21)}{\log(7)}$$

Calculate x using a calculator.

Using a calculator, compute the value of x: $$x \approx 1.77124$$ Therefore, the solution to the given equation is \(x \approx 1.77124\).

Key concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear as exponents. In this exercise, we work with the exponential equation \(7^x = 21\). The primary objective is to find the value of \(x\).
For these types of equations, a common strategy is to use logarithms to solve for the unknown variable. This is useful as logarithms help in bringing down the exponent for easier manipulation.
Whatever the base of the exponent is, like the 7 in our example, you aim to isolate the variable exponent by using properties of logarithms, particularly the logarithm power rule.

Logarithm Power Rule

The logarithm power rule is a crucial concept when solving exponential equations such as \(7^x = 21\). It states that \(\log(a^{b}) = b\log(a)\).
This rule helps in simplifying the equation by allowing us to move the exponent in front of the logarithm, effectively transforming the equation from an exponent problem to a linear one. In our example, we took the logarithm of each side to get \( \log(7^x) = \log(21) \).
- Using the logarithm power rule, this expression becomes \( x\log(7) = \log(21) \).
Once simplified, all that remains is to solve the linear equation for \(x\). This process is central to converting and solving exponential equations through logarithmic manipulation.

Solving Equations

Solving equations often requires isolating the variable you are solving for. In the context of our initial logarithmic equation, \( x\log(7) = \log(21) \), we aim to isolate \(x\).
To do this, you can perform operations that simplify the presence of \(x\). By dividing both sides of the equation by \(\log(7)\), you isolate \(x\), leading to the solution \( x = \frac{\log(21)}{\log(7)} \).
This expression can't be simplified further by hand to a nice number, so using a calculator for the final computation is necessary. Upon calculation, \( x \approx 1.77124 \).
- This approach highlights the importance of understanding the properties of logarithms in solving equations that appear complex initially.
Practicing these steps reinforces the ability to transform and solve equations of this kind efficiently.