Question

For question 19 Solve for a in the equation below. It may be helpful to convert the equation into exponential form.

log, 49 = x
x =

Answer 1

Question 19

`x = \boxed{\quad7\quad}`

Question 20

`\large y = \boxed{\quad 50\;e^(-2.3026t)\quad}`

Explanation:

Question 19

`\log_(7)49 = x\\\\`

Using the log rule

`\mathrm{If \; $log_a (y)= x$ then $y = x^a$ we get }\\\\49 = 7^x\\\\\mathrm{We \; know\;49 = 7^2}\\\\= > x = 2`

Question 20

`\mathrm{ Let\; y = 50 \cdot 0.1^t}`

Therefore

`(y)/(50) = 0.1^t\\\\\ln\left ((y)/(50)\right) = ln(0.1^t)`

Using the log rule

`\ln(a^t) = t \ln(a)\\\\\ln(0.1^t) = t \ln(0.1)\\\\\ln(0.1) = -2.3026\\\\= > \ln\left((y)/(50)\right) = -2.3026t\\\\\\`

using the log rule:
`\mathrm{ If \;\ln(y) = kt\;then\; y = e^(kt)}`
we get

`(y)/(50) = e^(-2.3026t)\\\\\mathrm{Multiply \;both \;sides \;by \;50}:\\\\y = \boxed{50e^(-2.3026t)}`