Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 2^{-x}=1.5 $$

Short answer

x = - \frac{\ln(1.5)}{\ln(2)}.

Step-by-step solution

Write the equation in logarithmic form

Given the equation is \(2^{-x} = 1.5\). To solve for \(-x\), take the natural logarithm (ln) on both sides of the equation: \(\ln(2^{-x}) = \ln(1.5)\)

Use the properties of logarithms

Apply the power rule of logarithms, which states \(\ln(a^b) = b \ln(a)\). Thus, \(\ln(2^{-x}) = -x \ln(2)\). Now, the equation becomes \(-x \ln(2) = \ln(1.5)\).

Solve for \(x\)

Isolate \(x\) by dividing both sides of the equation by \(- \ln(2)\). This yields: \(-x = \frac{\ln(1.5)}{\ln(2)}\). Hence, solving for \(x\), \(x = - \frac{\ln(1.5)}{\ln(2)}\).

Key concepts

Logarithmic Form

To solve exponential equations, the first step is to convert them into a different form called the logarithmic form. An exponential equation has the structure of \(a^{x} = b\). In our problem, the equation is given as \(2^{-x} = 1.5\). We need to express this using logarithms.
To do this correctly, apply the natural logarithm (ln) on both sides of the equation. This transforms the equation into a form that utilizes the logarithm: \(\ln(2^{-x}) = \ln(1.5)\). This step simplifies the process by allowing us to apply properties of logarithms.

Properties of Logarithms

Logarithms have several useful properties that help in solving equations. One key property is the power rule, which states that the logarithm of a number raised to a power is equal to the power times the logarithm of the base: \(\ln(a^{b}) = b \ln(a)\).
In our equation, after taking the natural logarithm, we have \(\ln(2^{-x}) = \ln(1.5)\). Applying the power rule, this becomes: \(-x \ln(2) = \ln(1.5)\).

Next, isolate the variable by dividing both sides by \(\ln(2)\):
\(-x = \frac{\ln(1.5)}{\ln(2)}\).
Finally, solve for \(x\) by multiplying both sides by \(-1\), resulting in \(x = - \frac{\ln(1.5)}{\ln(2)}\).

Natural Logarithm

The natural logarithm is a specific logarithm with the base of the mathematical constant \(e\), which is approximately equal to 2.71828. It is denoted as \(\ln\).

Natural logarithms are particularly useful when dealing with exponential functions, as they have properties that simplify complex equations. For instance, \(\ln(e) = 1\) and \(\ln(1) = 0\).

In our exercise, taking the natural logarithm of both sides converts the exponential form into a manageable equation that utilizes these properties: \(\ln(2^{-x}) = \ln(1.5)\). This step is crucial because it allows the application of logarithmic properties to isolate and solve for the variable.

By understanding natural logarithms and their properties, you can effectively solve a wide range of exponential equations.