Anonymous

# write the expression as a single logarithm and simplify?

log *b*3x+4(log*b*x-log*b*y)

*b*=subscript

Relevance

First look at log*b*x-log*b*y and simplify this. Since the operation is subtraction, you can combine this into a single logarithm through division. It equals log*b*(x/y). So now we have:

log*b*3x+4(log*b*(x/y))

The 4 becomes the exponent on (x/y), giving us

log*b*3x+log*b*(x/y)^4

Now, these two expressions can be combined into a single log function through multiplication since they are being added, giving us:

log*b*(3x)(x/y)^4.

We can simplify (3x)(x/y)^4 further. (x/y)^4=(x^4)/(y^4).

(3x)(x^4)=3x^5, so the final answer is:

log*b*((3x^5)/(y^4))

• you use formula

log*b* a +log*b* c = log*b* ac and

log*b* a -log*b* c = log*b* a/c and

log*b* a^c = c log *b*a

so the expression is

log*b*(3x *(x/y)^4)

log*b* (3x^5/y^4)

• All logs are to the base 'b' - I will just write log

log(3x) + 4(log x - log y)

=log(3x) + 4log(x/y) ----- log a - log b = log(a/b)

=log(3x) + log(x/y)^4 ---- nlog(a) = log(a^n)

=log(3x.(x/y)^4)) ------ log(a) + log(b) = log(ab)

=log(3x^5 / y^4)

• log *b* ((3x^5)/(y^4))