Logarithm

Log properties, equations

Logarithms are the inverse operation of exponentiation. While exponents ask 'what is the result of repeated multiplication?', logarithms ask 'how many times must we multiply to get this result?' The logarithm of a number to a given base is the exponent to which the base must be raised to produce that number. Logarithms simplify complex calculations involving multiplication, division, and powers, and are essential in many areas of mathematics, science, and engineering.

Key Formulas

\text{log\_b(a) = x means b\textasciicircum{}x = a (Definition of logarithm)}

\text{log\_b(1) = 0 for any base b > 0, b != 1}

\text{log\_b(b) = 1 for any base b > 0, b != 1}

\text{log\_b(a x c) = log\_b(a) + log\_b(c) (Product Rule)}

\text{log\_b(a/c) = log\_b(a) - log\_b(c) (Quotient Rule)}

\text{log\_b(a\textasciicircum{}n) = n x log\_b(a) (Power Rule)}

\text{log\_b(a) = log\_c(a) / log\_c(b) (Change of Base Formula)}

\text{log\_b(a) = 1 / log\_a(b) (Reciprocal Relation)}

\text{b\textasciicircum{}(log\_b(a)) = a (Inverse Property)}

\text{log\_b(a) x log\_a(b) = 1}

Key Concepts

Understanding Logarithm Notation

In log_b(a), 'b' is called the base and 'a' is called the argument or antilogarithm. Common bases include 10 (common logarithm, written as log or lg) and e ~= 2.718 (natural logarithm, written as ln). The base must be positive and not equal to 1, and the argument must be positive. Logarithms are only defined for positive real numbers.

Fundamental Logarithm Laws

The three fundamental laws are: (1) Product Rule - log of a product equals sum of logs, (2) Quotient Rule - log of a quotient equals difference of logs, and (3) Power Rule - log of a power equals exponent times log of base. These laws allow us to break down complex expressions into simpler components and are the foundation of all logarithmic calculations.

Change of Base Formula

The change of base formula log_b(a) = log_c(a) / log_c(b) is crucial when you need to evaluate logarithms with bases not available on standard calculators. Most calculators only compute log base 10 and log base e (ln). To find log_2(8), you would calculate ln(8)/ln(2) or log(8)/log(2), both giving 3. This formula is also useful for comparing logarithms with different bases.

Characteristic and Mantissa

For common logarithms (base 10), any positive number can be written in scientific notation as m x 10^n where 1 <= m < 10. The logarithm is then n + log(m), where n is the characteristic (integer part) and log(m) is the mantissa (decimal part, always positive). The characteristic of log(N) is: (number of digits in N) - 1 for N > 1, and negative for 0 < N < 1.

Antilogarithms

The antilogarithm is the inverse operation of logarithm. If log_b(x) = y, then antilog_b(y) = x = b^y. To find the antilog, you raise the base to the power of the given logarithm value. For common logarithms, antilog(x) = 10^x. For natural logarithms, antilog(x) = e^x. Antilogarithms are used to recover original numbers from their logarithmic values.

Solving Logarithmic Equations

To solve equations involving logarithms: (1) Use logarithm laws to simplify expressions, (2) Convert to exponential form when beneficial, (3) Always check that solutions satisfy the domain (argument > 0, base > 0, base != 1). For equations like log_b(x) = log_b(y), we get x = y. For log_b(x) = c, we get x = b^c. Multiple logarithmic terms often require combining them first.

Solved Examples

Problem 1:

Find the value of log_2(32) + log_3(27) - log_5(125).

Solution:

Step 1: Evaluate each logarithm separately.
Step 2: log_2(32) = log_2(2^5) = 5 (since 2^5 = 32).
Step 3: log_3(27) = log_3(3^3) = 3 (since 3^3 = 27).
Step 4: log_5(125) = log_5(5^3) = 3 (since 5^3 = 125).
Step 5: Calculate: 5 + 3 - 3 = 5.
Answer: 5

Problem 2:

If log_10(2) = 0.3010 and log_10(3) = 0.4771, find log_10(72).

Solution:

Step 1: Factor 72 into prime factors: 72 = 8 x 9 = 2^3 x 3^2.
Step 2: Apply product rule: log_10(72) = log_10(2^3 x 3^2) = log_10(2^3) + log_10(3^2).
Step 3: Apply power rule: = 3 x log_10(2) + 2 x log_10(3).
Step 4: Substitute given values: = 3(0.3010) + 2(0.4771).
Step 5: Calculate: = 0.9030 + 0.9542 = 1.8572.
Answer: 1.8572

Problem 3:

Simplify: log_5(125) x log_8(64) / log_10(1000).

Solution:

Step 1: Evaluate each logarithm.
Step 2: log_5(125) = log_5(5^3) = 3.
Step 3: log_8(64) = log_8(8^2) = 2 (since 8^2 = 64).
Step 4: log_10(1000) = log_10(10^3) = 3.
Step 5: Calculate: (3 x 2) / 3 = 6/3 = 2.
Answer: 2

Problem 4:

If log(x + y) = log x + log y, find the value of x in terms of y.

Solution:

Step 1: Apply the product rule in reverse: log x + log y = log(xy).
Step 2: So log(x + y) = log(xy).
Step 3: Since logarithms are one-to-one functions: x + y = xy.
Step 4: Rearrange: xy - x = y.
Step 5: Factor: x(y - 1) = y.
Step 6: Solve for x: x = y/(y - 1).
Answer: x = y/(y - 1)

Tips & Tricks

Always verify the domain before solving: the argument must be positive and the base must be positive and not equal to 1.
Memorize common logarithm values: log_2(2)=1, log_2(4)=2, log_2(8)=3, log_2(16)=4, log_2(32)=5, log_3(9)=2, log_5(25)=2, log_10(100)=2.
When given log values like log(2) or log(3), look for ways to express the target number as products or powers of these known values.
For equations with logarithms of different bases, use the change of base formula to convert to a common base.
If log_b(x) = 0, then x = 1 (regardless of the base, as long as it's valid).
When simplifying complex logarithmic expressions, work from the inside out - simplify the arguments first before applying log rules.

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