determine whether an expression is correct using rules of natural log.

Question

Determine if each is correct or incorrect

(a) ln(4) + ln(x) − ln(y) = ln(4x)/ ln(y) 

I said this one was correct because adding is multiplication, and subtracting is dividing. So I think its correct.


(b) 10^(7log(x)) = 7x
 For this one I am less sure, I think it might be correct because the 10 and the log terms may cancel, but again, I am not sure.

(c) (log(x))^4 = 4log(x)

This one I have no idea, I think it may be correct because exponents become coefficients but the parenthesis are throwing me off. what do they mean?

Answer 1

(a) ln(4) + ln(x) − ln(y) = ln(4x)/ ln(y)   False!

I said this one was correct because adding is multiplication, and subtracting is dividing. YOUR STATEMENT IS WRONG.

 

The left side can be combined as a single logarithm: ln[4x/y]

 

(b) 10^(7log(x)) = x⁷  so their claim is False

 

(c) (log(x))^4 = 4log(x)  False

Answer 2

Properties of logs:  (1). log_ax + log_ay = log_a(xy)

 

                           (2). log_ax - log_ay = log_a(x/y)

 

                           (3). log_a(x^r) = r(log_ax)

 

                           (4). a^log_ax = x

 

(a).  ln4 + lnx - lny = ln(4x) - lny  (by property 1)

         

                            = ln(4x/y) (by property 2)

 

       This is not the same as ln(4x)/lny.  So, (a) is false.

 

(b). 10^7logx = 10^7log₁₀x (Remember that logx is an abbreviation for                                            log₁₀x)

 

                = 10^{log₁₀(x^7)} (by property 3)

 

                = x⁷ (by property 4)

 

        This is not equivalent to 7x, so (b) is also false.

 

(c). (logx)⁴ = (logx)(logx)(logx)(logx), which is not 4logx

 

         (c) is false as well. 

 

      Note:  4logx = log(x⁴)  (by property (3))