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Exponential and Logarithmic Equations

Exponential and Logarithmic Equations. 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 b x = x ln b or ln e x = x . 4. Solve for the variable.

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Exponential and Logarithmic Equations

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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.

  4. Solve: 54x – 7 – 3 = 10 Text Example cont. Solution 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.

  5. 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}. ? ? 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 Text Example 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.

  6. 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

  7. 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

  8. Exponential and Logarithmic Equations

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