SOLUTION: Solve each equation. Show all steps and work. Round your answers to four decimal places. a. 6^2x=250 b.〖log〗_3 (x-3)+〖log〗_3 (x-7)=4 Given the function f(x)=〖

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Solve each equation. Show all steps and work. Round your answers to four decimal places. a. 6^2x=250 b.〖log〗_3 (x-3)+〖log〗_3 (x-7)=4 Given the function f(x)=〖      Log On


   

Question 1138686: Solve each equation. Show all steps and work. Round your answers to four decimal places.
a. 6^2x=250
b.〖log〗_3 (x-3)+〖log〗_3 (x-7)=4


Given the function f(x)=〖log〗_2 (x-2)-1.
1. Graph the function
2. Find its domain & range
3. Write the equation of its asymptote
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

a.
6%5E%282x%29=250....take a log of both sides
log%286%5E%282x%29%29=log%28250%29
2x%2Alog%286%29=log%282%2A5%5E3%29............since log%286%29=log%282%2A3%29=log%282%29%2Blog%283%29, and log%282%2A5%5E3%29=log%282%29+%2B+3log%285%29, we have
2x%28log%282%29%2Blog%283%29%29=log%282%29+%2B+3log%285%29
x=%28log%282%29+%2B+3log%285%29%29%2F2%28log%282%29%2Blog%283%29%29
x=%280.69314718+%2B+4.8283137373%29%2F2%280.69314718%2B1.098612288668%29
x=5.5214609173%2F3.583518937336
x=1.5408

b.
log%283%2C+%28x-3%29%29%2Blog%283+%2C%28x-7%29%29=4 .....convert to base 10
log%28%28x-3%29%29%2Flog%283%29%2Blog%28%28x-7%29%29%2Flog%283%29=4 ....both sides multiply by log%283%29
log%28%28x-3%29%29%2Blog%28%28x-7%29%29=4+log%283%29
log%28%28x-3%29%28x-7%29%29=+log%283%5E4%29.......since log same, we have
%28x-3%29%28x-7%29=+3%5E4
x%5E2-7x-3x%2B21=+81
x%5E2-10x%2B21-+81=0
x%5E2-10x-+60=0......use quadratic formula
x=%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F2a
x=%28-%28-10%29%2B-sqrt%28%28-10%29%5E2-4%2A1%2A%28-60%29%29%29%2F%282%2A1%29
x=%2810%2B-sqrt%28100%2B240%29%29%2F2
x=%2810%2B-sqrt%28340%29%29%2F2
x=%2810%2B-18.4391%29%2F2
x=%285%2B-9.2195%29
solutions:
x=5%2B9.2195=>x=14.2195
x=5-9.2195=>x=+-4.2195=> since log disregard negative solution
so, your solution is:x=14.2195


Given the function f%28x%29=+log%282%2C+%28x-2%29%29-1+
1. Graph the function
+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+log%282%2C+%28x-2%29%29-1%29+
2. Find its domain & range
domain: { x element R : x%3E2 }
range: R (all real numbers)
3. Write the equation of its asymptote
Vertical asymptote: x=2
log%28x+-+2%29%2Flog%282%29+-+1->-infinity as x-%3E2


Horizontal asymptote: none+