How to use the properties of logarithms to rewrite the expression as a single logarithm and simplify....

log (base 7)(x-4)+log (base 7)(x+4)=

Using Log(base 7)(a) + Log (base7)(b) = Log (base 7)(ab):

Log (base 7)(x-4) + Log (base 7)(x+4)= Log (base 7) ((x+4)(x-4))

= Log (base 7)(x^2 + 4x - 4x + 16) = Log (base7)(x^2 - 16)

Hi Jessica James:

log (base 7)(x-4)+log (base 7)(x+4) = 0 - original equation and setting y to 0

log(base 7) ( ( x-4)(x+4)) = 0 - Pulling out the log(base 7)

7^(log(base 7) ( ( x-4)(x+4))) = 7^0 anti-log both sides of it

(x-4)(x+4) = 1 - Solving the anti-log of both sides

x^2 - 16 = 1 - Foil

x^2 -16 + 16 = 16 + 1 - Adding the additive inverse of a number to both sides of the equation to move it to the other side of it

x^2 = 17 -Addition

x = +/- sq rt(17) - Square rooting both sides of the equation to remove the square and adding the ( + / - ) because this equation as two answers


x = - 4.123 105 625 618 - solving the minus side of the exact answer

or

x = 4.123105625618 - solving the Plus side of the exact answer

proof or check

log (base 7)(x-4)+log (base 7)(x+4)= 0 - original equation and setting y to 0

log(x-4)/log(7) + log(x+4) / log(7) = 0 - Change of base rule of logarithms

log(4.123105625618 - 4) / log(7) + log(4.123105625618 + 4) / log(7) = 0 - plugging x with 4.123105625618

log(4.123105625618 - 4) / 0.845098040014 + log(4.123105625618 + 4) / 0.845098040014 = 0 - plugging the log(7) with 0.845098040014

log(0.123105625618) / 0.845098040014 + log (8.123105625618)/ 0.845098040014 = 0 - Addition & subtraction of 4.123105625618 with 4 and -4

(-0.909722100449/ 0.845098040014) + (0.909722100449/ 0.845098040014) - solving the logs of 0.123105625618 and 8.123105625618

-1.076469305779 + 1.076469305779 = 0 - Division

0 = 0 - Addition

it checks and equals