MTH 115 - Review for Test 3 - Szalankiewicz

Solve. log x 64 = 3

3√x^3 = 3√64, x = 4

Rewrite the expression as the logarithm of a single quantity. 1/2 log10 s + 5 log10 q

= log10 s↑1/2 + log10 q↑5 = log10 √s + log10 q↑5 = log10 [(√s)(q↑5)]

Determine the antilog in base 10 of the given number. Round to four decimal places. 1.1033

10^1.1033 = 12.6853

Find the indicated value. Find P if log P = 4, where P is the population of bacteria in a culture.

10^4 = P = 10,000

log3 q + log3 r

= log3 (q×r)

log6 7 - log6 z

= log6 (7/z)

4 loga x - loga y

= loga x↑4 - loga y = loga (x↑4/y)

Find the exact value of the expression. log5 1/5

5^x = 1/5 = 5X-1 = -1

Find the exact value of the expression. log6 1/36

6^x = 1/36 = 1/6^2 = 6^-2 = -2 (6x6 = 36, 6^-2 = 1/36)

(Solve each problem) The formula A = -3(log 10 Pi - log 10 Po) gives the power gain (in dB) if Pi is the input and Po is the output. Write the expression on the right side as the logarithm of a single quantity.

A = -3 (log↓10 Pi - log↓10 Po) = -3 log(Pi/Po). A = log(Pi/Po)↑-3

For a particular radioactive sample, the decay rate C in disintegrations per second can be modeled by the equation logeC = loge244 - 0.25t, where t is the elapsed time in minutes. Solve this equation for the decay rate C as a function of time t.

Log↓eC = f(t); Log↓eC also means ln. lnC = ln244 - .25t; subtract ln244 from both sides. ln(C/244) = -.25t → exponential form is e↑-.25t = C/244. C = 244e↑-.25t

(Solve each problem) The height in meters of women of a certain tribe is approximated by h = 0.52 + 2 log (t/3) where t is the woman's age in years and 1≤t 20. Estimate the height (to the nearest hundredth) of a woman of the tribe 4 years of age. (Round to the nearest hundredth.)

h = .52 + 2 log(4/3) = .52 + 2(.1249287366) = .77m

(Solve each problem) If interest is compounded continuously (daily compounded interest closely approximates this), with an interest rate i (expressed as a decimal), a bank account will double in t years according to i = (ln 2)/t. Find i if the account is to double in 6.5 years.

i = (ln2)/6.5 = 0.1066

Express the equation in exponential form. log^1/2 8 = -3

is (1/2)^-3 = 8

Express the equation in exponential form. log4 1/16 = -2

is 4^-2 = 1/16

Express the equation in exponential form. log5 25 = 2

is 5^2 = 25

Write as a sum or difference of logarithms. Your results should not contain any exponents or radicals. log4 6x

is log4 6 + log4 x

Write as a sum or difference of logarithms. Your results should not contain any exponents or radicals. log6 xy

is log6 x + log6 y

Write as a sum or difference of logarithms. Your results should not contain any exponents or radicals. log8 3/5

is log8 3 - log8 5

Determine the natural logarithm of the number. Round to the nearest ten-thousandth. 0.000243

ln(0.000243) = -8.3324

(Solve each problem) Under certain conditions, the electric current i in a circuit containing a resistance R and an inductance L is given by ln i/I = -Rt/L, where t is the time (in seconds), and I is the current at t = 0. Calculate how long (in seconds) it takes the current to reach 0.600 of the initial value, if I = 0.618 A, R = 6.40 Ω, and L = 1.23 H.

ln0.600 = (-6.40t/1.23). Divide 1.23/-6.40 from both sides. (1.23/-6.40)ln0.600 = -6.40/ln0.600 then multiply by 1.23. t = .10 seconds

Determine the logarithm in base 10 of the given number to four decimal places. 0.715

log(0.715)= -0.1456939582

Determine the value of the logarithm. Round to the nearest hundredth. log4 0.75

log(o.75)/log(4) or ln(0.75)/ln(4) = -0.21

Convert to logarithmic form. 2^-3 = 1/8

log2 (1/8) = -3

log3 3 + log3 x

log3 (3x)

Convert to logarithmic form. 3^5 = 243

log3 243 = 5

Write as a sum or difference of logarithms. Your results should not contain any exponents or radicals. log8 √10/13

log8 √10 - log8 13 = 1/2 log8 √10 - log8 13

Convert to logarithmic form. 9^1/2 = 3

log9 3 = 1/2

Evaluate the exponential function at the specified value of x. y = 2^x, x = -8

y = 2^-8 = 1/2^8 = 1/256

Evaluate the exponential function at the specified value of x. y = 4^x, x = 3

y = 4^3 = 64

Solve for y in terms of x. logb y = logb 5 + logb x

y = 5x

Solve for y in terms of x. logb y = logb 9 - logb x

y = 9/x

(Solve each problem) The distance x traveled by a motorboat in t seconds after the engine is cut off is given by x = k↑-1 ln (kv0t + 1), where v0 is the velocity of the boat at the time the engine is cut and k is constant. Find how long it takes a boat to go 210 m if v0 = 18.0 m/s and k = 6.70 × 10↑-3/m.

240 = (6.70 × 10↑-3)↑-1 ln[6.70 × 10↑-3)(18)t + 1. Divide (6.70 × 10↑-3)↑-1 from both sides. 1.407 = ln[0.1206t + 1] = e↑1.407 = 0.1206t + . Subtract 1 from both sides. Divide 0.1206 from both sides. t = 25.6 seconds.

(Solve each problem) Scientists compare intensity levels of two sounds by taking their difference: S2 - S1 = 10 log (I2/Io) - 10 log (I1/Io). Write the expression on the right side as the logarithm of a single quantity.

S↓1/S↓2 = 10 [log(l↓2/l↓0) - log(l↓1/l↓0)] =10 [log(1↓2/l↓1)] S↓1 - S↓2 = log(l↓2/l↓1)^10

(Solve each problem) Open parallel transmission lines have a characteristic impedance given by Zo = 240(log 10 a - log 10 b). Write the expression on the right side as the logarithm of a single quantity.

Zo = 240(log a/b) Zo = log (a/b)↑240

Determine the antilog, base e, of the given number. Round to the nearest ten-thousandth. 2.5177

e^2.5177 = 12.4000