Question

if log10x=a and log10y=b express log10sqrt(10x/y^3) in terms of a and b

Answer 1

\[\log_{10} x=a \]\[\log_{10} y=b\]\[x=10^a\]\[y=10^b\]
\[\log_{10} \sqrt{10*10^a/(10^b)^3}\]

Answer 2

simplify using exponent rules

Answer 3

\[\log_{10} \sqrt{100a}/\sqrt{1000b^3}\]

Answer 4

Should we multiply the denominator by sqrt of 1000B^3

Answer 5

\[\log_{10} \sqrt{100^a*1000^*(-3b)}\]

Answer 6

\[\log_{10} \sqrt{10*10^a/(10^b)^3}\]

Answer 7

\[\large \log_{10} \sqrt{10^{a+1}/10^{3b}}\]

Answer 8

I dont Understand why is 10^a+1 and 10^3b instead of 100^a and 1000^3b

Answer 9

thats a good question, so you're asking why \(10*10^{a}\) becomes \(10^{a+1}\) right ?

Answer 10

yes

Answer 11

lets try this :

\(10^3 = 10 *10*10\)
\(10^4 = 10 *10*10 * 10\)

\(\cdots \)

\(10^a = 10*10*10* \cdots a~ times\)

right ?

Answer 12

if we multiply 10 again, clearly we need to increase the exponent :

\(10^{a+1} = 10 * (10*10*10*\cdots a~times )\)
\(~~~~~~~~~~ = 10* 10^a \)

Answer 13

see if that looks more or less convincing

Answer 14

We can generalize this rule :

\(\large x^m * y^n = x^{m+n} \)

Answer 15

\(\large 10*10^a\)

\(\large 10^1*10^a\)

\(\large 10^{a+1}\)

Answer 16

iam convinced now by the last part thank you so much @ganeshie8 :)

Answer 17

good to hear :)