Exams · Russian state exam, profile level

No. 7 Values of expressions

Find the value of an expression with logarithms, powers or trigonometry. No calculator: everything cancels by the rules.

How to solve it

Expressions with logarithms

No calculator is needed here: an expression like $\log_5 65 - \log_5 13$ folds into a single number by the rules of logarithms. The job is to see which rule applies.

Step by step

Look at the bases. If they are equal, a sum of logarithms becomes the logarithm of a product, and a difference the logarithm of a quotient: $\log_5 65 - \log_5 13 = \log_5 5 = 1$.
A number in front of a logarithm can move inside as an exponent: $2\log_3 5 = \log_3 25$.
An expression of the form $a^{\log_a b}$ equals $b$.
What usually remains at the end is $\log_a a^k = k$.

The first argument.

The second argument. Both logarithms have the same base.

Example: $\log_6 4 + \log_6 9 = \log_6 36 = 2$.

The same base below and in the logarithm.

The number you get. By definition $\log_a b$ is the exponent that turns $a$ into $b$.

Example: $3^{\log_3 7 + 1} = 3^{\log_3 7} \cdot 3 = 21$.

Common mistakes

Adding logarithms with different bases, although the rule works only for equal ones.
Splitting the logarithm of a sum: $\log_a(x + y)$ is not $\log_a x + \log_a y$.
Confusing a difference of logarithms with their quotient: $\log_a x - \log_a y = \log_a \frac{x}{y}$, not $\frac{\log_a x}{\log_a y}$.

Example

Expressions with powers

An expression like $\frac{2^5 \cdot 4^3}{8^3}$ looks heavy, but once every number is written as a power of two it folds into a single power. No big multiplications needed.

Step by step

Bring all the powers to one base: $4 = 2^2$, $8 = 2^3$, $\frac{1}{2} = 2^{-1}$.
When multiplying powers add the exponents, when dividing subtract them, when raising a power to a power multiply them.
Work out the final exponent and compute the power.

The exponent of the first power.

The exponent of the second power. The rules hold only when the bases are equal.

Example: $\frac{2^5 \cdot 4^3}{8^3} = \frac{2^5 \cdot 2^6}{2^9} = 2^{5 + 6 - 9} = 2^2 = 4$.

Common mistakes

Multiplying the exponents when multiplying powers: $2^3 \cdot 2^4 = 2^7$, not $2^{12}$.
Merging powers with different bases: $2^3 \cdot 3^2$ is not a single power.
Confusing a negative exponent with a negative number: $2^{-3} = \frac{1}{8}$, not $-8$.

Example

Trigonometric expressions

You need a value like $10\cos 420^\circ$. The angle is large, but sine and cosine repeat every full turn, $360^\circ$, and the reduction formulas bring any angle down to an acute one from the table.

Step by step

Remove full turns: $\cos 420^\circ = \cos(420^\circ - 360^\circ) = \cos 60^\circ$.
For a negative angle use symmetry: $\cos(-\alpha) = \cos \alpha$, $\sin(-\alpha) = -\sin \alpha$.
Bring the angle down to an acute one with the reduction formulas: $\sin(180^\circ - \alpha) = \sin \alpha$, $\cos(180^\circ + \alpha) = -\cos \alpha$.
Take the value from the table: $\sin 30^\circ = \frac{1}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\sin 60^\circ = \frac{\sqrt{3}}{2}$.

With $180^\circ$ the function stays the same. The sign is that of the original function in the quadrant where the angle lies.

With $90^\circ$ and $270^\circ$ sine turns into cosine and back. The sign is found the same way, from the quadrant.

Example: $\sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2}$.

Common mistakes

Forgetting the sign: cosine is negative in the second and third quadrants, sine in the third and fourth.
Swapping sine and cosine where it is not needed, with $180^\circ$ and $360^\circ$.
Losing the minus of a negative angle in a sine: $\sin(-30^\circ) = -\frac{1}{2}$.