Solving Exponential Equations - Part 2 of 2

Steps to Solve Exponential Equations 00:11

"We isolate the exponential part, take the logarithm of both sides, and solve for the variable."

• The initial approach to solving exponential equations involves isolating the exponential term. If there are two exponential parts, they should be placed on opposite sides of the equation.

• Either the common logarithm (base 10) or the natural logarithm (base e) can be used since these are commonly available on calculators.

• After applying the logarithm, the next step is to solve for the variable and subsequently check the solutions visually using graphs.

Solving the First Example 00:35

"Taking the natural logarithm allows us to apply the power property to simplify our equation."

• For the first example, the goal is to isolate (7^x), resulting in the equation (7^x = 5) after adding one to both sides.

• Since it is not possible to change the base to a common number, the natural logarithm is applied to both sides: (\ln(7^x) = \ln(5)).

• The power property of logarithms converts the exponent into a coefficient, allowing the equation to be rewritten as (x \cdot \ln(7) = \ln(5)).

• Dividing both sides by (\ln(7)) enables us to find that (x) is approximately (0.83).

Alternative Method to Find Exponential Solutions 01:38

"Rewriting the equation in a logarithmic form offers another valid pathway to the solution."

• An alternative method shown involves rewriting the equation (7^x = 5) in logarithmic form as (\log_7(5) = x).

• Utilizing the change of base formula provides the same expression for (x) that was derived earlier: (\frac{\ln(5)}{\ln(7)}).

Checking Solutions Graphically 02:24

"Using a graphing calculator, we can verify our solution by finding the intersection point of the two functions."

• To confirm the accuracy of the solution, the left side of the original equation is input as (y_1) and the right side as (y_2) into a graphing calculator.

• Adjustments to the viewing window may be necessary, and the intersection point's x-coordinate validates the computed value of (x).

Solving a More Complex Example with Two Exponential Parts 03:21

"When dealing with two exponential parts, we take logarithms of both sides to balance the equation."

• In the next example with two exponential functions, (3 \times 2^x = 15), the exponential part is isolated first.

• The logarithm is applied producing (\ln(2^x) = \ln(5)). After simplifying, we find (x) from (\frac{\ln(5)}{\ln(2)}), yielding an approximate result of (2.32).

Handling Two Exponential Parts in One Equation 04:32

"Using logarithmic identities helps to manipulate equations involving multiple exponential terms."

• When faced with the equation ( \left( \frac{2}{3} \right)^x = 5^{3-x} ), the logarithm is taken on both sides.

• This results in a more complex equation where rearranging and factoring allows for solving (x) by combining terms involving (x) and isolating it.

• The computed value of (x) is confirmed through graphing the relevant functions.

Final Complex Problem with Variables on Both Sides 07:10

"Comparing the coefficients of like terms helps to isolate and solve for the variable efficiently."

• Another equation involving both sides showcases exponents with terms on one side.

• After isolating and balancing the constants versus the variable terms, (x) is determined again by estimating using a graphing calculator.

• Verifying with the original equation ensures accuracy, illustrating a hands-on approach for confirming mathematical solutions.