Warmup Solve 2x – 4 = 14. Round to the nearest ten-thousandth. Show all logarithmic work.

6-8 Natural Logarithms Evaluate expressions with natural base and natural logarithms. Solve exponential equations and inequalities using natural logarithms.

𝑒 1 = ? 𝐴 𝐶𝑂𝑁𝑆𝑇𝐴𝑁𝑇!

Write an equivalent exponential or logarithmic function 20. ln 0.25 =𝑥

Write an equivalent exponential or logarithmic function 22. 𝑒 𝑥−3 =2

Write as a single logarithm Write as a single logarithm. End result should have no operations and no powers. 29. 7 ln 1 2 +5𝑙𝑛 2

Solve. Round to the nearest ten-thousandth. 37. (isolate the exponential part, then put in logarithmic form to get x by itself)

Solve each inequality. Round to the nearest ten-thousandth. 40. 𝑒 𝑥 ≥42.1

Recall that if interest is compounded n times a year, we use 𝐴=𝑃 (1+ 𝑟 𝑛 ) 𝑛𝑡 If the interest is compounded every second of every day, or continuously, we use: Exponential: 𝐴=𝑃 𝑒 𝑟𝑡 Logarithmic: 𝑙𝑛 𝐴 𝑃 =𝑟𝑡

51. Use the logarithmic and exponential formulas for continuous interest You are on your own for a, b. c) Double money in 9 years, what rate? d) 10,000 after 12 yrs at 4.75%, what is initial deposit?