The value of [tex]\rm{log\ 10^{19}[/tex] is 19.
Logarithm
A log function is a way to find how much a number must be raised in order to get the desired number.
[tex]a^c=b[/tex]
can be written as
[tex]\rm{log_ab=c[/tex]
where a is the base to which the power is to be raised,
b is the desired number that we want when power is to be raised,
c is the power that must be raised to a to get b.
For example, let's assume we need to raise the power for 10 to make it 1000 in this case log will help us to know that the power must be raised by 3.
[tex]\rm{log\ 1000 = 3[/tex]
Given to us
log a = 3,
log b = 4,
log c = -1,
Logarithm Values
log a = 3
[tex]10^3 = a\\1000 = a\\a=1000[/tex]
lob b = 4,
[tex]10^4=b\\10000=b\\b=10,000[/tex]
log c = -1,
[tex]10^{-1} = c\\\\\dfrac{1}{10} = c\\\\c = 0.1[/tex]
Simplifying
[tex]\rm{log\dfrac{b^7}{a^5c^6}\\[/tex]
[tex]=\rm{log\dfrac{10,000^7}{1,000^5\times 0.1^6}\\[/tex]
[tex]=\rm{log 10^{19}[/tex]
Hence, the value of [tex]\rm{log\ 10^{19}[/tex] is 19.
