(a)
x = log 5/12
<soapbox> Careful: PEMDAS has not much of a place in Algebra, certainly not in Geometry, Trigonometry, or Calculus… imho… so, log(5/12), rather than log(5)/12, yes ?… Parentheses please ! Sorry, I (and others) feel that PEMDAS is “junk Math”, but that’s an ignorable opinion. The best part of PEMDAS is to use the “P” and be done with the rest. </soapbox>
x = log₁₀(5/12) = log₁₀(5) - log₁₀(12) = log₁₀(5) - log₁₀(3*2²) = log₁₀(5) - [ log₁₀(3) + 2*log₁₀(2) ]
= log₁₀(5) - log₁₀(3) - 2*log₁₀(2) = -0.380211
[Check: 10ˣ = 10^(-0.380211) ] = 0.4166… = 5/12
[Alert, Alert !! Some Calculators have quirky behavior here... if you do the above, to obtain log(x) = -0.380211, then do 10ˣ to go back to 5/12, you may instead get: -10^(0.380211) = -2.4 rather than 5/12 = 0.41666.... Solution: Make sure you have the Parentheses around the signed exponent (use Parens ! Where have we heard that before?): 10^(-0.380211). Yes, this should be considered a "bug" in iOS Calculator [I am a Software Engineer of 40 years, worked at Apple a long time, and I will report this. ]
(b)
x = "5 square root (72.43)* 3 square root (.00035)^2 all divided by six square root 5.84"
x = [ 5√(72.43) * 3√(.00035)² ] / 6√(5.84) = [ 5(72.43)¹/² * 3[(.00035)²]¹/² ] / 6(5.84)¹/²
[ Note for later CHECK: x = .00308149 => log(x) = -2.511239 ]
x = [ 5(72.43)¹/² * 3(.00035) ] / 6(5.84)¹/² ==>
log(x) = [ log(5) + (1/2)log(72.43) + log(3) + log(.00035) ] - [ (log(6) + (1/2)log(5.84) ]
= log(5) + log(3) + log(.00035) - log(6) + (1/2)[ log(72.43) - log(5.84) ]
= log(5) + log(3) + log(.00035) - log(2*3) + (1/2)[ log(72.43) - log(5.84) ]
= log(5) + log(3) + log(.00035) - log(2) - log(3) + (1/2)[ log(72.43) - log(5.84) ] # log(3) cancellation
= log(5) + log(35*10⁻⁵) - log(2) + (1/2)[ log(72.43) - log(5.84) ]
= log(5) + log(5*7*10⁻⁵) - log(2) + (1/2)[ log(72.43) - log(5.84) ]
= log(5) + log(5) + log(7) - 5 - log(2) + (1/2)[ log(72.43) - log(5.84) ]
= [ 2*log(5) + log(7) - 5 - log(2) ] + (1/2)[ log(72.43) - log(5.84) ]
Now, we can go more crazy with this exercise, as follows:
= [ 2*log(5) + log(7) - 5 - log(2) ] + (1/2)[ log(7243*10⁻²) - log(584*10⁻²) ]
= [ 2*log(5) + log(7) - 5 - log(2) ] + (1/2)[ -2 + log(7243) - (-2 + log(584)) ]
= [ 2*log(5) + log(7) - 5 - log(2) ] + (1/2)[ log(7243) - log(2³*73) ]
= [ 2*log(5) + log(7) - 5 - log(2) ] + (1/2)[ log(7243) - 3*log(2) - log(73) ] # 73 and 7243 are prime
[Check] log(x) = -3.0579919 + .5467528 ==> log(x) = -2.511239 [Check: see above]
x = 10ˡᵒᵍ⁽ˣ⁾ = .00308149
(c)
2.35ˣ = 342.7 ==> log(2.35ˣ) = log(342.7) ==> x*log(2.35) = log(342.7) # log(aˣ) = x*log(a)
x = log(342.7) / log(2.35) = 6.831403
[Check] 2.35^(6.831403) = 342.7
