Using Exponentials and Logs

Using Exponentials and Logs

Basics of Exponentials

• Recall that any number, say ‘a’, raised to the power of ‘0’ is 1, i.e., a^0 = 1.
• Understand that any number, say ‘a’, raised to the power ‘1’ is simply ‘a’, i.e., a^1 = a.
• Recognize that the power rule states that the power of a power can be calculated by multiplying the exponents, i.e., (a^m)^n = a^(m*n).

Basics of Logarithms

• Remember that logarithms are the inverse of exponentials, i.e., log_a(b) = n implies a^n = b.
• Understand that the log of any number to the same base is always 1, log_a(a) = 1.
• Know the rule: log_a(1) = 0.

Properties of Exponential and Logarithmic Functions

• Exponential functions are always positive, i.e., a^x > 0 for any real ‘x’.
• An exponential function raised to an increasing power increases if a>1 and decreases if 0<a<1.
• Logarithmic functions are undefined for non-positive numbers, i.e., log_a(x) is undefined for x ≤ 0.
• Logarithmic functions are decreasing when a between 0 and 1, and increasing when a is greater than 1.

Manipulating Exponentials and Logarithms

• Learn the identities to simplify expressions involving exponentials and logarithms.
• The power rule: a^(m+n) = a^m * a^n.
• The change of base formula: log_a(b) = log_c(b)/log_c(a).
• The product rule in logarithms: log_a(m*n) = log_a(m) + log_a(n).
• The quotient rule in logarithms: log_a(m/n) = log_a(m) - log_a(n).

Remember that being comfortable with these rules, properties, and being able to manipulate expressions are key to master exponentials and logarithms. Practice solving a variety of problems to get a strong grip over these topics.