Question

Solve each exponential equation. Express irrational solutions in exact form. $$ \left(\frac{3}{5}\right)^{x}=7^{1-x} $$

Short answer

x = \frac{\text{ln}(7)}{\text{ln}(\frac{3}{5}) + \text{ln}(7)}

Step-by-step solution

Apply logarithms to both sides

Start by taking the natural logarithm (ln) of both sides of the equation to make use of the properties of logarithms:\[ \text{ln}\bigg(\bigg(\frac{3}{5}\bigg)^{x}\bigg) = \text{ln}\bigg(7^{1 - x}\bigg) \]

Use the power rule of logarithms

Apply the power rule of logarithms which states that \( \text{ln}(a^b) = b \text{ln}(a) \):\[ x \text{ln}\bigg(\frac{3}{5}\bigg) = (1 - x) \text{ln}(7) \]

Distribute the logarithm on the right side

Distribute the natural logarithm on the right hand side:\[ x \text{ln}\bigg(\frac{3}{5}\bigg) = \text{ln}(7) - x \text{ln}(7) \]

Combine like terms

Combine all the terms involving \( x \) and move non-variable terms to the other side:\[ x \text{ln}\bigg(\frac{3}{5}\bigg) + x \text{ln}(7) = \text{ln}(7) \]Factor out \( x \) from the left side:\[ x \bigg( \text{ln}\bigg(\frac{3}{5}\bigg) + \text{ln}(7) \bigg) = \text{ln}(7) \]

Solve for x

Isolate \( x \) by dividing both sides of the equation by \( \text{ln}\bigg(\frac{3}{5}\bigg) + \text{ln}(7) \):\[ x = \frac{\text{ln}(7)}{\text{ln}\bigg(\frac{3}{5}\bigg) + \text{ln}(7)} \]

Key concepts

natural logarithm

The natural logarithm, often abbreviated as ln, is a logarithm to the base e, where e is an irrational constant approximately equal to 2.71828. It is important in mathematics because it simplifies many complex exponential functions. In our exercise, using the natural logarithm helps us transform an exponential equation into a linear one, making it easier to solve. We start by applying the natural logarithm to both sides of the original exponential equation. This step is crucial because it allows us to use the properties of logarithms to manipulate and simplify the equation.
The equation \(\text{ln}\bigg(\bigg(\frac{3}{5}\bigg)^{x}\bigg) = \text{ln}\bigg(7^{1 - x}\bigg)\) is key in transforming the problem into a form we can solve.

properties of logarithms

Logarithms have several important properties that make them useful for solving equations like the one in our exercise. Here are a few crucial properties:

• The product property: \( \text{ln}(a \times b) = \text{ln}(a) + \text{ln}(b) \)
• The quotient property: \( \text{ln}\bigg(\frac{a}{b}\bigg) = \text{ln}(a) - \text{ln}(b) \)
• The power property: \( \text{ln}(a^b) = b \text{ln}(a) \)

In our problem, we use the power property extensively. This property allows us to bring the exponent in an exponential expression down as a coefficient, simplifying the equation. Applying \( \text{ln}\bigg(\bigg(\frac{3}{5}\bigg)^{x}\bigg) = x \text{ln}\bigg(\frac{3}{5}\bigg)\) is a direct application of the power property of logarithms.
This manipulation is essential for the steps that follow.

power rule of logarithms

The power rule of logarithms is a vital tool in simplifying the expressions we are working with. According to this rule, \( \text{ln}(a^b) = b \text{ln}(a) \). In our exercise, we apply this rule to both sides of the equation after taking the natural logarithm. For example, \( \text{ln}\bigg(\bigg(\frac{3}{5}\bigg)^{x}\bigg) = x \text{ln}\bigg(\frac{3}{5}\bigg)\)
This transformation changes the nature of the problem, turning complex exponentials into manageable linear terms. Similarly, on the right side, we have:
\( \text{ln}(7^{1-x}) = (1-x) \text{ln}(7) \)
This simplification helps us set up the equation for combining like terms and solving for the variable x straightforwardly.

combining like terms

Combining like terms is a fundamental step in solving algebraic equations. Once the equation has been log-transformed and simplified using logarithmic properties, we end up with terms involving the variable x on both sides. Our goal is to collect these terms together to isolate x. From our exercise, we have:
\( x \text{ln}\bigg(\frac{3}{5}\bigg) + x \text{ln}(7) = \text{ln}(7)\)
We combine the x terms by factoring x out:
\( x \bigg( \text{ln}\bigg(\frac{3}{5}\bigg) + \text{ln}(7) \bigg) = \text{ln}(7) \)
Now, it's simple to isolate x by dividing both sides by \(\text{ln}\bigg(\frac{3}{5}\bigg) + \text{ln}(7)\). Thus, we get:
\( x = \frac{\text{ln}(7)}{\text{ln}\bigg(\frac{3}{5}\bigg) + \text{ln}(7)} \)
This result provides x in its exact form, showcasing the power of logarithmic properties in solving exponential equations.