{{ article.displayTitle }}

Now use the definition of a logarithm. This helps to remove the variable from the exponent. Finally, the equation can be solved for Just to make sure the obtained solution is not an extraneous solution, will be substituted in the equation. Since a true statement was obtained, is the solution to the equation. This means that the chapati will become unsafe to eat due to fungi growth about hours after cooking it.

To solve the inequality, all numbers involved will first be written as powers of To start, will be written as Now, all the numbers can be written as powers of The base of the expressions on both sides is which is is greater than Therefore, by the Exponential Property of Inequality, the left-hand side of the inequality is less than the right-hand side if and only if the exponent on the left-hand side is less than the exponent on the right-hand side. The inequality can finally be solved.

To find the hydrogen ion concentration of a solution with a pH of the number must be substituted for pH in the given formula. Next, to solve the equation, each side will be considered as a function. The first function is a horizontal line whose intercept is The second is a logarithmic function. Considering that its domain is make a table of values.

Now, graph the functions on the same coordinate plane.

The graphs intersect at one point. The coordinate of the point of intersection is the hydrogen ion concentration of the solution.

The exact value of the coordinate cannot be identified. However, the answer must be rounded to the nearest mole per liter, so the hydrogen concentration of the solution with a pH of is rounded to the nearest mole per liter. The last step is to check the solution by substituting it back into the equation. Since a true statement was obtained, is a solution to the equation.

The equation involves natural logarithms. To solve it, the first step is to rewrite the right-hand side as a single logarithm by using the Power Property of Logarithms. Since both sides are single logarithms with the same base, the Property of Equality for Logarithmic Equations can be used. Notice that the resulting equation is a quadratic equation. Therefore, the values of and need to be identified. Next, and will be substituted in the Quadratic Formula. Finally, the solutions can be verified by substituting them into the original equation. The solution will be checked first. Logarithms are only defined for positive numbers, so is an extraneous solution. The solution will be checked now. A true statement was found this time. Therefore, the only solution to the equation is

To calculate the watts per square meter of a typical rock concert, will be substituted for in the given equation. The logarithmic equation will then be solved algebraically. Recall that For simplicity, the units will not be considered until the end. Next, to eliminate the logarithm, its definition will be used. Since the base is not stated, the formula involves a common logarithm. Finally, the equation can be solved for A rock concert with a decibel level is has a sound with an intensity of watts per square meter. To make sure that this is not an extraneous solution, it can be confirmed by substituting the value back into the equation. Since a true statement was obtained, is a solution to the equation.

To solve the inequality, the first step is writing the right-hand side as a single logarithm with base Start by writing as a logarithmic expression by using the definition of a logarithm. With this information, can be substituted for Then, the Quotient Property of Logarithms can be used. Both sides are written as single logarithms with base which is greater than Therefore, by the Logarithmic Property of Inequality, the left-hand side of the inequality is less than the right-hand side if an only if the argument of the left-hand side is less than the argument of the right-hand side. Now solve this new inequality by using inverse operations. Finally, recall that logarithms are only defined for positive numbers. Therefore, the arguments of the logarithms in the original inequality must be greater than zero.

Argument on the LHS	Argument on the RHS

The solution set is the intersection of three inequalities. To find the solution set, the inequalities can be graphed on the same number line. The solution set consists of the numbers on the line that belong to the three individual solutions. In other words, the solution set is the intersection of all three solutions. The solution to the inequality are all real numbers greater than and less than because the three lines overlap between and This solution can be written as a compound inequality or in interval notation.

Since the expressions have different bases, the Property of Equality for Exponential Equations cannot be used. However, both and are greater than zero for any value of Therefore, common logarithms can be taken on both sides. Now use the Power Property of Logarithms. Then the variable can be isolated. The solution to the exponential equation is This can verified by substituting it into the original equation.