Question 704768
The part "correct to 4 significant figures" must have come from general instructions given for a test including many questions.
The instructions may have included using logarithms as needed, but logarithms are not needed to solve that equation.
{{{4=32(1-2x)}}} --> {{{4=32*1-32*2x}}} applying the distributive property
{{{4=32*1-32*2x}}} --> {{{4=32-64x}}} performing indicated multiplications
{{{4=32-64x}}} --> {{{4+64x-4=32-64x+64x-4}}} adding {{{64x-4}}} to both sides of the equal sign
{{{4+64x-4=32-64x+64x-4}}} --> {{{64x+4-4=32-64x+64x-4}}} --> {{{64x=32-4}}} --> {{{64x=28}}}
rearranging (commutative and associative properties) and doing indicated operations
The explanation for that step could get even more involved, if you wanted
{{{64x=28}}} --> {{{64x/64=28/64}}} --> {{{x=28/64}}} --> {{{x=7/8}}}
I do not need a calculator, or a slide rule, or logarithms to divide 7 by 8 to find the answer as a decimal.
I can calculate that in my head, without even paper and pencil, thinking this
{{{7/8=1-1/8=1-1*125/(8*125)=1-125/1000=1-0.1250000=0.8750000}}}
 
USING LOGARITHMS:
I have not needed to use logarithms to divide since calculators were invented.
(I hope you are not expected to do that)
If it was a more complicated fraction/division, and I only had a table of base 10 logarithms, paper, and pencil, I would calculate {{{7/8}}} as a decimal correct to 4 significant figures like this:
{{{log(7/8)=log(7) - log (8)=0.8450980-0.9030900=-0.0579920=-1+1-0.0579920=-1+0.9420080}}}
Then, I would look in the table for the number that has {{{0.9420080}}} for a logarithm.
I would see that
{{{log(8.74)=0.9415114}}}
{{{log(8.75)=0.9420081}}}
{{{log(8.76)=0.9425041}}}
Since the difference between consecutive logs is {{{log(8.75)-log(8.74)=0.0004967}}} for numbers differing in {{{0.01}}},
if would not even try to interpolate to get a more accurate answer,
because I would figure (without even calculating it), that
{{{log(8.749)=0.9420081-0.01*0.000497=0.9420081-(0.001/0.01)*0.0000497=0.9419584}}}
is much farther than {{{log(8.750)}}} from {{{0.9420080}}}.
So, {{{highlight(x=0.8750)}}} is an answer correct to 4 significant figures.