 Section: Algebra INDEX Introduction Aims Exponential equations Logarithms

EXPONENTIAL EQUATIONS, LOGARITHMS AND SIMULTANEOUS EQUATIONS

INTRODUCTION

Exponential and logarithmic functions are the most common types of functions which exist in the world around us. As a result there are many problems which may require the use of exponential or logarithmic equations in order to solve them.

One example of this is the Richter scale, which measures the magnitude M of an earthquake according to the amplitude of its surface waves A Hence: M=log A+C where C =3.3+1.66 logD-logT is a constant which depends on the period of time that the waves are registered on the seismograph T and the distance from the epicenter in angular degrees D. If we want to calculate the amplitude (intensity) of the seismic wave we would need to solve a logarithmic equation.

We would also need to solve equations if we wanted to find the necessary time in hours (t) for the amount of bacteria Escherichia coli, found in the intestinal tract of many mammals, to reach a certain number.

(P=P0.2t/D  where P= 8000 bacteria,  P=500 D=30).

In the same way, if we wanted to work out the age of a bone found at an archaeological dig, and we knew that it contained 20% of carbon 14, which is present in all animal life, we would need to solve the equation: 0.2=e-0.000121t .

AIMS
• To distinguish between and solve individual and simultaneous exponential equations.

• To identify the decimal solutions of exponential equations with logarithms.

• To distinguish between and solve individual and simultaneous logarithmic equations.

• To understand how exponential and logarithmic equations relate to each other.

 Leoncio Santos Cuervo Spanish Ministry of Education. Year 2001 