Logarithm

• Exponential Function is the function defined as 𝑏𝑦𝑓 (𝑥) = 𝑏𝑥 where 𝑏 > 0, 𝑏 ≠ 1.
• In exponential function, the variables are always in the exponent.
• Exponential equation can be solved by applying several laws of exponent.
• Exponential inequality is like exponential equation but is used in either of the inequality form: <, >, ≤, ≥, instead of =.
• Exponent of One is represented as ��^1 = ��^6.
• Exponent of Zero is represented as 𝑥^0 = 1.
• Exponent Product is represented as 𝑥^𝑚 * 𝑥^𝑛 = 𝑥^𝑚 + 𝑛𝑥^2 * 𝑥^3.
• 𝒙-intercepts are where functions cross the 𝑥-axis.
• In the case of a logarithmic function, its domain is defined as a set of all positive real numbers (𝑥 > 0) while its range is a set of real numbers (− ∞, ∞).
• An asymptote is a line that a curve approaches (but never touches), as it heads towards infinity.
• An intercept is where a function crosses the x and y axis.
• They are found algebraically by setting 𝑦 = 0 and solving for 𝑥.
• A vertical asymptote is when as 𝑥 approaches some constant value 𝑐 (either from left to right), then the curve goes towards ∞ or − ∞.
• They are also called roots, solutions, and zeroes of a function.
• The zero of a function is the 𝑥-value that makes the function equal to 0, that is, 𝑓 (𝑥) = 0.
• Logarithmic inequalities (with inequality symbols) are inequalities in which one (or both) sides contain a logarithm.
• A logarithmic function is expressed as 𝑦 = 𝑙𝑜𝑔 𝑏 𝑎 , wherein y is the exponent of the exponential function, b is the base and a is the answer ( or argument) when b is raised to exponent c.
• Logarithmic equation (with equal sign) is an equation in which one (or both) sides contains a logarithm.
• Common Logarithms are logarithms with base 10; 𝑙𝑜𝑔 𝑥 is a short notation for 𝑙𝑜𝑔 10 𝑥.
• The logarithm of a product of two numbers is the same as adding two logarithms.
• Natural Logarithms are logarithms to the base 𝑒 (approximately 2.71828 ), denoted by 𝑙𝑛 𝑥.
• The logarithm of the quotient of two numbers is the same as subtracting two logarithms.
• Exponent Quotient is represented as 𝑥^𝑚 𝑥^𝑛 = 𝑥^𝑚 − 𝑛𝑥^6 𝑥^2.
• Exponent Power (𝑥^𝑚)𝑛 = 𝑥^𝑚 * 𝑛 (𝑥^2)³.
• Exponent Power of a Product (𝑥 * 𝑦)𝑛 = 𝑥𝑛 * 𝑦𝑛 (𝑥𝑦)³.
• Exponent Power of a Quotient (𝑥𝑦)𝑛 = 𝑥𝑛𝑦𝑛 (𝑥𝑦)².
• Exponent Negative (𝑥 − 𝑛) = 1𝑥𝑛 (𝑥 − 3) = 1𝑥³.
• The logarithm of a with base 𝑏 is denoted by 𝑙𝑜𝑔 𝑏 𝑎 and is defined as 𝑐 = ��𝑜𝑔 𝑏 𝑎 if and only if 𝑎 = 𝑏 𝑐𝒂 = 𝒃 𝒄 ⇔ 𝒄 = 𝒍𝒐𝒈 𝒃 𝒂.
• In both the logarithmic and exponential forms, b is the base.
• Logarithmic and exponential functions are inverses.
• If 𝑏 ˃ 1 , then the exponential function 𝑦 = 𝑏 𝑥 is increasing for all 𝑥.
• The value of 𝑙𝑜𝑔 𝑏 𝑥 can be negative.
• In the logarithmic form 𝑙𝑜𝑔 𝑏 𝑥 , 𝑥 cannot be negative ( 𝑥 > 0 )
• If 0 < 𝑏 > 1 , then the exponential function 𝑦 = 𝑏 𝑥 is decreasing for all 𝑥.
• If both sides of an inequality are multiplied by or divided by the same positive real number, the sense of the inequality is not changed.
• If the same real number is added to or subtracted from both sides of an inequality, the sense of the inequality is not changed.
• Let 𝑎 , 𝑏 , and 𝑐 be positive real numbers such that �� ≠ 1.
• If both sides of an inequality are multiplied by or divided by the same negative real number, the sense of the inequality is changed.
• Logarithm is the exponent or power to which a base must be raised to yield a given number.
• In the exponential form, c is an exponent; this implies that the logarithm is actually an exponent.