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Solving Exponential Equations using Natural Logarithms

Learn how to isolate exponential expressions and solve equations utilizing natural logarithms step-by-step. Practice problems and solutions included.

richardfranklin
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Solving Exponential Equations using Natural Logarithms

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  1. Exponential and Logarithmic Equations

  2. 1. Isolate the exponential expression. 2. Take the natural logarithm on both sides of the equation. 3. Simplify using one of the following properties: ln bx = x ln b or ln ex = x. 4. Solve for the variable. Using Natural Logarithms to Solve Exponential Equations

  3. Solve: 54x – 7 – 3 = 10 Text Example Solution We begin by adding 3 to both sides to isolate the exponential expression, 54x – 7. Then we take the natural logarithm on both sides of the equation. 54x – 7 – 3 = 10 This is the given equation. 54x – 7 = 13 Add 3 to both sides. ln 54x – 7 = ln 13 Take the natural logarithm on both sides. (4x – 7) ln 5 = ln 13 Use the power rule to bring the exponent to the front. 4x ln 5 – 7 ln 5 = ln 13 Use the distributive property on the left side of the equation. 4x ln 5 = ln 13 + 7 ln 5 Isolate the variable term by adding 7 ln 5 to both sides. x = (ln 13)/(4 ln 5) + (7 ln 5)/(4 ln 5) Isolate x by dividing both sides by 4 ln 5. The solution set is {(ln 13 + 7 ln 5)/(4 ln 5)} approximately 2.15.

  4. Check log4 (x + 3) = 2 This is the logarithmic equation. log4 (13 + 3) = 2 Substitute 13 for x. log4 16 = 2 2 = 2 This true statement indicates that the solution set is {13}. ? ? Text Example Solve: log4(x + 3) = 2. Solution We first rewrite the equation as an equivalent equation in exponential form using the fact that logbx = c means bc = x. log4 (x + 3) = 2 means 42 = x + 3 Now we solve the equivalent equation for x. 42 = x + 3 This is the equivalent equation. 16 = x + 3 Square 4. 13 = xSubtract 3 from both sides.

  5. Solve 3x+2-7 = 27 Solution: 3x+2= 34 ln 3 x+2 = ln 34 (x+2) ln 3 = ln 34 x+2 = (ln 34)/(ln 3) x+2 = 3.21 x = 1.21 Example

  6. Solve log 2 (3x-1) = 18 Solution: 2 18 = 3x-1 262,144 = 3x - 1 262,145 = 3x 262,145 / 3 = x x = 87,381.67 Example

  7. How long will it take 25,000 to grow to 500,000 at 9% annual interest compounded monthly?

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