Write each expression as the logarithm of a single quantity: log3 x + log3 (x+2) - log3 8?

Question

The 3s are lowered. Assume that x,y,z and b are positive numbers and b=/ 1. Use the properties of logarithms to write each expression as the logarithm of a single quantity. Please show work and give a full explaination so will know how to do this in the future.

Answer 1

log3 x + log3 (x+2) - log 3 (8)
= log3 (x) (x + 2) - log3 (8)
= log3 (x) (x + 2)/ 8

Answer 2

x = 10 or -10 log3(x^2+8) - log3(4) = log3((x^2+8)/4) = log3((x^2)/4+2), you can do this by using the rule that log(x)-log(y) = log(x/y) (also log(x) + log(y) = log(x*y)) so log3((x^2)/4+2) = 3, since it's log base 3 take both sides of the equation to the 3rd power, this cancels out the log3. Then you get (x^2)/4 +2 = 27 => (x^2)/4 = 25 => x^2 = 100 => x = 10 or -10

Answer 3

log3 ( [x(x+2)] / 8 ) =
log3 ( [x² + 2x] / 8 )

When you add two logs of the same base you multiply the terms:
log x + log y = log(xy) since both logs have base of 10.

When you subtract two logs of the same base you divide the terms:
log x - log y = log(x/y) since both logs have base of 10.

So in this problem you have the form:
log x + log y - log z.
So you have log ( (xy/z) )

Answer 4

In the following "log" means log to base 3
log x + log (x + 2) - log 8
= log [ x.(x + 2) / 8 ]

Answer 5

log x + log y =log xy
log x - log y =log x/y


log(base3) x + log(base3) (x+2) - log(base3) 8
= log(base3)(x(x+2)/8)
=log(base3)((x^2+2x)/8)

Answer 6

log(base3)((x^2+2x)/8)