Logarithm is the inverse function of exponentiation.
It helps us solve equations involving exponents.
Properties of logarithms simplify complex logarithmic expressions.
We can apply these properties to manipulate and solve logarithmic equations.
Let’s review the properties of logarithms.
Property 1: The logarithm of a product is equal to the sum of the logarithms.
Equation: log(ab) = log(a) + log(b)
Example: log(2*3) = log(2) + log(3)
Property 2: The logarithm of a quotient is equal to the difference of the logarithms.
Equation: log(a/b) = log(a) - log(b)
Example: log(8/2) = log(8) - log(2)
Property 3: The logarithm of a power is equal to the exponent multiplied by the logarithm.
Equation: log(a^b) = b * log(a)
Example: log(4^3) = 3 * log(4)
Property 4: The logarithm of the base is always 1.
Equation: log(b) = 1
Example: log(10) = 1
Property 5: The logarithm of 1 is always 0.
Equation: log(1) = 0
Example: log(1) = 0
Property 6: The logarithm of a negative number or zero is undefined.
Equation: log(x), x <= 0 → undefined
Example: log(-1) = undefined
Property 7: The logarithm of a negative number or zero is undefined.
Equation: log(x), x <= 1 → undefined
Example: log(0.5) = undefined
Property 8: The logarithm of the base itself is always 1.
Equation: log(b, b) = 1
Example: log(2, 2) = 1
Property 9: The logarithm of a number with a base greater than 1 is always positive.
Equation: log(b, x), b > 1 → positive
Example: log(5, 100) = positive
Property 10: The logarithm of a number with a base between 0 and 1 is always negative.
Equation: log(b, x), 0 < b < 1 → negative
Example: log(0.5, 100) = negative
Property 11: The logarithm of a number raised to the power of the logarithm’s base is equal to the logarithm itself.
Equation: log(b, b^x) = x
Example: log(2, 2^3) = 3
Property 12: The logarithm of a number to the base 10 is commonly known as the common logarithm.
Equation: log(x) = log10(x)
Example: log(100) = log10(100)
Property 13: The logarithm of a number to the base e, where e is Euler’s number, is known as the natural logarithm.
Equation: log(x) = ln(x)
Example: log(2) = ln(2)
Property 14: The logarithm of a number to a base greater than 1 yields a positive value.
Equation: log(b, x), b > 1 → positive
Example: log(3, 10) = positive
Property 15: The logarithm of a number to a base between 0 and 1 yields a negative value.
Equation: log(b, x), 0 < b < 1 → negative
Example: log(0.5, 10) = negative
Property 16: Changing the base of the logarithm can be done using the formula:
Equation: log(b2, x) = log(b1, x) / log(b1, b2)
Example: log(2, 100) = log(10, 100) / log(10, 2)
Property 17: The logarithm of a number that is a power of the base is equal to the exponent.
Equation: log(b, b^x) = x
Example: log(3, 3^4) = 4
Property 18: The logarithm of a number that is 1 less than the base is equal to zero.
Equation: log(b, b-1) = 0
Example: log(5, 4) = 0
Property 19: The logarithm of a number that is one more than the base is equal to 1.
Equation: log(b, b+1) = 1
Example: log(7, 8) = 1
Property 20: The logarithm of a number that is a fraction between the base is between 0 and 1.
Equation: log(b, b^x) = x
Example: log(0.5, 0.5^2) = 2
Property 21: The logarithm of 1 to any base is always 0.
Property 22: The logarithm of a base raised to a logarithm is the logarithm itself.
Property 23: The logarithm of the reciprocal of a number is equal to the negative logarithm of the original number.
Property 24: The logarithm of a product raised to a power can be split into the log of each factor raised to that power.
Property 25: The logarithm with a base equal to 1 is undefined.
Property 26: The exponential function and logarithmic function are inverse functions of each other.
Property 27: The logarithm of a number greater than 1 is always positive.
Property 28: The logarithm of a number between 0 and 1 is always negative.
Property 29: The logarithm with a base equal to 0 is undefined.
Property 30: The logarithm of a negative number is undefined in the real numbers.