MCAT Physics and Math Review
Chapter 10: Mathematics
10.2 Exponents and Logarithms
For many students, exponents and logarithms are topics filed away in the depths of memory. While exponential and logarithmic functions are uncommon in everyday life, a number of science topics and equations regularly tested on the MCAT require use of these concepts, as shown in Table 10.1.
Topic 
Equation 
Location in Kaplan MCAT Review Series 
Sound level 
Chapter 7 of MCAT Physics and Math Review 

Exponential decay 
n = n_{0}e^{−}λt 
Chapter 9 of MCAT Physics and Math Review 
Arrhenius equation for activation energy 
Chapter 5 of MCAT General Chemistry Review 

Gibbs free energy 
ΔG°_{rxn} = −RT ln K_{eq} 
Chapter 7 of MCAT General Chemistry Review 
p scales (pH, pOH, pK_{a}, pK_{b}) 
pH = −log [H+] 
Chapter 10 of MCAT General Chemistry Review 
Henderson–Hasselbalch equation 
Chapter 10 of MCAT General Chemistry Review 

Table 10.1. Common Exponential and Logarithmic Equations on the MCAT 
EXPONENTS
In addition to exponential equations, exponents appear frequently on the MCAT in the context of scientific notation, discussed above. Here, we look at the rules of arithmetic with exponents.
Exponent Identities
Only a basic understanding of exponents is necessary for the MCAT, although it can be helpful to know a few values and basic rules. First, any number to the zeroth power is equal to 1:
X^{0} = 1
Equation 10.1
When adding or subtracting numbers with exponents, the true value must be calculated before the addition or subtraction can be performed. For example, 3^{2} + 3^{2} ≠ 6^{2}; rather, 3^{2} + 3^{2} = 9 + 9 = 18. However, if the base and exponent are the same, we can add the coefficients: 3^{2} + 3^{2} = (1 + 1) × 3^{2} = 2 × 3^{2} = 18.
In cases of multiplication and division, the exponents can be manipulated directly, as long as the base number is the same. When multiplying two numbers with the same base, the exponents are added to determine the new number:
X^{A} × X^{B} = X^{(}A + B^{)}
Equation 10.2
KEY CONCEPT
When adding, subtracting, multiplying, or dividing numbers with exponents, the base must be the same.
In division, we subtract the exponent of the denominator from the exponent in the numerator to find the exponent in the quotient, as long as all bases are the same:
Equation 10.3
For a number that is raised to an exponent and then raised again to another exponent, the two exponents are multiplied:
(XA^{)}B = X^{(}A × B^{)}
Equation 10.4
When a fraction is raised to an exponent, the exponent is distributed to the numerator and denominator:
Equation 10.5
Negative exponents represent inverse functions:
Equation 10.6
For fractional exponents, the numerator can be treated as the exponent, and the denominator represents the root of the number:
Equation 10.7
Estimating Square Roots
On Test Day, you may be expected to calculate approximate square roots. To do so, it is useful to be familiar with the values in Table 10.2.
X 
X^{2} 
X 
X^{2} 
X 
X^{2} 
X 
X^{2} 
1 
1 
6 
36 
11 
121 
16 
256 
2 
4 
7 
49 
12 
144 
17 
289 
3 
9 
8 
64 
13 
169 
18 
324 
4 
16 
9 
81 
14 
196 
19 
361 
5 
25 
10 
100 
15 
225 
20 
400 
Table 10.2. Square Values of Integers from 1 to 20 
If you are asked to calculate the square root of any number less than 400, you can approximate its value by determining which two perfect squares it falls between. As an alternative method, you can divide the number given to you by known squares to attempt to reduce it:
One can estimate this value by considering that the square root of five is somewhere between 2 and 3 (2^{2} = 4 and 3^{2} = 9), and is closer to 2 than 3. If we estimate to be about 2.2, then which is congruent with our knowledge that the square root of 180 will be between 13 and 14. The true value of is approximately 13.42.
If you are using a number in scientific notation, adjust the decimal by one place if necessary so that the exponent is easily divisible by two:
MCAT EXPERTISE
Estimation of square roots and logarithms is generally sufficient to the first decimal place; don’t struggle to become more precise because it won’t be necessary on Test Day.
Finally, it is useful to know the values of and :
Equation 10.8
RULES OF LOGARITHMS
Logarithms follow many of the same rules as exponents because they are inverse functions. The logarithmic rules are described below:
Equations 10.9 to 10.14
It is also useful to know that “p” can be shorthand for −log; thus, pH = −log [H^{+}], pK_{a} = −log K_{a}, and so on.
Example:
Derive the Henderson–Hasselbalch equation from the expression for K_{a}.
Solution:
COMMON vs. NATURAL LOGARITHMS
Logarithms can use any base, but the most common are base ten, as in our decimal system, and base e (Euler’s number, about 2.718). Baseten logarithms (log_{10}) are called common logarithms, whereas those based on Euler’s number (loge or ln) are called natural logarithms. Both common and natural logarithms obey the rules discussed above, but it can be easier to estimate common logarithms because of our familiarity with the decimal number system. Therefore, it is useful to be able to convert between natural logarithms and common logarithms:
Equation 10.15
KEY CONCEPT
e is Euler’s number, which is 2.718281828459045… . It is also the base for the natural logarithm.
Estimating Logarithms
When estimating the logarithm of a number, use scientific notation. An exact logarithmic calculation of a number that is not an integer power of 10 is unnecessary on the MCAT. The testmakers are interested, however, in testing your ability to apply mathematical concepts appropriately in solving certain problems. Fortunately, there is a simple method of approximation that can be used on Test Day. If a value is written in proper scientific notation, it will be in the form n × 10m, where n is a number between 1 and 10. From this fact, we can use logarithm rules to approximate the value:
Because n is a number between 1 and 10, its logarithm will be a decimal between 0 and 1 (log 1 = 0 and log 10 = 1). The closer n is to 1, the closer log n will be to 0; the closer n is to 10, the closer log n will be to 1. As a reasonable approximation, one can say that
log (n × 10m) ≈ m + 0.n
Equation 10.16
where 0.n represents sliding the decimal point of n one position to the left (dividing n by ten). For example, log (9.2 × 10^{8}) ≈ 8 + 0.92 = 8.92 (actual = 8.96).
BRIDGE
A similar concept for estimating logarithms is used in calculations of pH, as described in Chapter 10 of MCAT General Chemistry Review. The shortcut is slightly different because we are working with negative logarithms and a negative exponent in the case of pH:
−log (n × 10^{−}m) ≈ m − 0.n.
MCAT Concept Check 10.2:
Before you move on, assess your understanding of the material with these questions.
1. Simplify the following expressions:
· (a + b)^{2} =
·
· log_{a} (a) =
· log (a^{3}) − log (a) =
2. Estimate
3. Estimate log 7,426,135,420: