Mental Logs

Question 1

How do you “undo” a log function?

Also known as the antilog function

Answer 1

When, log(x) = y
use x = 10^y to undo the logarithm

All logarithm problems on the MCAT are base 10

Fundamentally, you are undoing the log function:

(your starting function): log(x) = y
(Raise both sides by the log’s base): 10^log(x) = 10^y
(Becuase 10^log(x) is simply x): x = 10^y

Question 2

What is the equation for a basic log function?

Answer 2

log₁₀(x) = y

All logarithm problems on the MCAT are base 10

Question 3

What is log(10)=?

Answer 3

1

We can verify this by performing the antilog of our answer:

10¹ = 10

Question 4

What is log(100)=?

Answer 4

2

We can verify this by performing the antilog of our answer:

10² = 100

Question 5

What is log(1000)=?

Answer 5

3

We can verify this by performing the antilog of our answer:

10³ = 1000

Question 6

What is log(1/10)=?

Answer 6

-1

We can verify this by performing the antilog of our answer:

10^-1 = 1/10 (becuase negative exponents place the base^x in the denominator so that base^-x = 1/base^x

Question 7

What is log(1/100)=?

Answer 7

-2

We can verify this by performing the antilog of our answer:

10^-2 = 1/100 (becuase negative exponents place the 10^x in the denominator so that 10^-x = 1/10^x

Question 8

What is log(1/1000)=?

Answer 8

-3

We can verify this by performing the antilog of our answer:

10^-3 = 1/10000 (becuase negative exponents place the base^x in the denominator so that base^-x = 1/base^x

Question 9

What is log(0.1/1)=?

Answer 9

-1, this is something you might see in a weak acid/base problem. However, do not let the fraction confuse you, this is simply log(1/10) but not in a reduced form.

Question 10

What is the general rule for log(x) where x<1?

Answer 10

log(x) when x<1 will always result in a negative value

Question 11

Write the expanded form of log(x/y)

Answer 11

log(x/y)=log(x)-log(y)

Question 12

What is log(2)=?

Memorization Required

Answer 12

log(2) is an irrational number, but we estimate it to be ~0.3 to use on the MCAT

Question 13

What is log(3)=?

Answer 13

log(3) is an irrational number, but we estimate it to be ~0.5 to use on the MCAT

Question 14

What is log(5)=?

Answer 14

log(5) is also an irrational number, but we estimate it to be ~0.7 for mental math on the MCAT

Question 15

What is log(0.2)=?

Will have to use memorized logs and expanded form to solve mentally

Answer 15

log(0.2) =-0.69897

Becuase log(0.2) =log(1/5)
we can expand this log, so that: log(1/5) = log(1) - log(5)
Since we memorized log(5), we can solve from here:
log(1)-log(0.5) = 0 - 0.7

Becuase log₁₀(1) = 0
confirm by performing antilog:
10⁰ = 1

Question 16

lnWhat is the general formula for a natural log (ln)?

Answer 16

The general formula for natural logarithms is the inverse of the natural base (e) exponential function. Therefore:

ln(x) = y

ln(x) is the same as log_e(x)

Question 17

How do you undo a natural log (ln) function?

Answer 17

When ln(x) = y, we use e^x=y to undo the logarithmic function

Follows the same rules as base ten logarithms

Question 18

What is log(200)=?

Will have to use memorized logs and expanded form to solve mentally

Answer 18

log(200) = 2.30103

Becuase log(200) = log(100) + log(2)
and we know that log(100) = 2 and log(2) = ~0.3
Therefore, log(100) + log(2) = 2 + 0.3 = ~2.3

Question 19

Write the expanded form of log(xy)

Answer 19

log(xy) = log(x) + log(y)

Similar rules are applied to trig functions